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in @xcite , sloane proposed the following curious problem . take a non - negative integer , write down its decimal representation , and multiply its digits together , getting a new non - negative integer . repeat the process until a single - digit number is obtained . the problem can thus be stated : is the number of steps taken in this process uniformly bounded ? let us start with a general formulation of sloane s problem , while at the same time introducing some of the notation that we will use . given a natural number @xmath1 , and an _ integer base _ @xmath2 , consider the base-@xmath3 expansion of the number @xmath1 , say @xmath4_q \;=\ ; \sum_{j=1}^k d_j q^{k - j}\ , \ ] ] where each digit @xmath5 ( and @xmath6 when @xmath7 ) . let @xmath8 denote the product of all such digits , _ i.e. _ , @xmath9 thus @xmath10 defines a map @xmath11 , which we call the _ sloane map in base @xmath3_. clearly , such map can be iterated : write @xmath8 in base @xmath3 , multiply its digits to obtain @xmath12 , and so on . in particular , given any @xmath13 we can consider its _ orbit _ under the sloane map , namely @xmath14 the following proposition ensures that this sequence always stabilizes after a finite number of steps .    [ trivialstart ] we have @xmath15 for all @xmath16 ( i.e. , as long as the base @xmath3 expansion of @xmath1 has at least two digits ) . write @xmath1 in base @xmath3 as in , and note that @xmath17 . since @xmath18 for all @xmath19 , it follows that @xmath20    from proposition [ trivialstart ] we deduce that @xmath1 is a fixed point of @xmath21 if and only if @xmath22 . it also follows from proposition [ trivialstart ] that every orbit of @xmath21 is finite and converges to some @xmath23 that is a fixed point . in other words , there exists a minimum number @xmath24 such that @xmath25 for all @xmath26 . hence @xmath24 _ is the smallest number @xmath27 such that @xmath28 has a single digit_. sloane asked in @xcite whether such minimum number of steps until a fixed point is uniformly bounded . the number @xmath24 is known as the _ persistence _ of @xmath1 in base @xmath3 . numerical evidence that @xmath24 is bounded has been collected for some values of @xmath3 . furthermore , the answer to sloane s question is trivially positive for @xmath29 since for any @xmath30 one has @xmath31 , and @xmath32 is the fixed - point set of @xmath33 . the problem  known as the _ persistence problem _  can be stated as follows . [ prob1 ] for a given @xmath34 , is there a positive number @xmath35 such that @xmath36 for all @xmath37 ? a related set of issues goes as follows ( considering now @xmath38 as an element of @xmath39 ) . [ prob2 ] what is the behavior of @xmath35 seen as a function of @xmath3 ? more precisely , one can ask :    1 . is the answer to problem [ prob1 ] positive for all , or all but finitely many , or most , or infinitely many , or perhaps only finitely many values of @xmath3 ? 2 .   what is the asymptotic behavior of @xmath35 as @xmath40 ?    here are some known facts about the persistence problem in various bases :    1 .   in base @xmath29 , the situation is rather trivial : every positive integer has persistence @xmath41 in base @xmath42 . 2 .   in base @xmath43 , no number with persistence greater than @xmath44 has ever been found . 3 .   in base @xmath45 , the number @xmath46 has persistence @xmath47 , because under the sloane map @xmath48 we have @xmath49 in fact , this is the smallest number with persistence equal to @xmath47 . 4 .   still in base @xmath45 , the number @xmath50 has persistence @xmath51 . it is the smallest number with persistence equal to @xmath51 . it is conjectured that @xmath52 for all @xmath1 . this has been checked for all @xmath1 up to @xmath53 .      in this paper , we have two main goals . the first goal is to examine the persistence problem in the light of some dynamical systems considerations . we will show that sloane s question ( problem [ prob1 ] ) has an affirmative answer in a certain _ probabilistic _ sense . roughly speaking , we will show that _ for any base @xmath3 , the set of natural numbers @xmath1 with persistence @xmath54 , _ i.e. _ such that @xmath55 , is an extremely rarified subset of @xmath56_. the probabilistic sense in question will be made progressively clear in  [ sec : erg1 ] and  [ sec : erg2 ] . we will see in particular that problem [ prob1 ] has a positive answer for @xmath43 if a precise orbit that we will fully describe has a `` generic '' behavior under the @xmath57-action determined by a well - defined piecewise affine degree one circle map . similarly , problem [ prob1 ] has a positive answer for @xmath58 if two precise orbits that we will fully describe have a `` generic '' behavior under the @xmath57-action determined by another well - defined piecewise affine degree one circle map . the affine circle maps that we will encounter here are defined by @xmath3 and a number @xmath59 , a digit in base @xmath3 .    for bases @xmath60 , the relevant dynamical systems for the sloane map are no longer @xmath57-actions , but rather @xmath61-actions with @xmath17 . more precisely , they are given by certain free abelian groups of piecewise affine degree - one circle maps . we will exploit some simple ergodic properties of such free - abelian actions in order to derive our main probabilistic result on the sloane map , namely theorem [ rankkdensity ] . our second goal is to formulate a very general conjecture , namely conjecture [ conj : convvtoequi ] , concerning the asymptotic distribution of digits in the base @xmath3 expansion of long products whose factors are chosen from a given finite set of primes . this conjecture is conveniently formulated in terms of certain objects that we call _ multiplication automata _ , in part because their time evolution produces patterns that resemble those produced by the evolution of ( one - dimensional ) cellular automata . the most general form of the conjecture , as stated in  [ multaut ] , is too broad to allow thorough numerical tests . thus , short of a proof , we felt the need to provide an heuristic explanation lending support to this conjecture . such heuristics is given in the language of _ inhomeogeneous markov chains _ , as a form of convergence to equilibrium for the evolution of our automata . the multiplication automata and the associated convergence to fair distribution may be of interest to the physics of growth processes , and perhaps to other aspects of statistical mechanics .      in order to avoid possible misunderstandings , we warn the reader that another map has been studied , inspired by @xmath21 : let us call it the erds - sloane map , and denote it by @xmath62 . for each natural number @xmath1 , @xmath63 is the product of the non - zero digits of @xmath1 in base @xmath3 . according to guy @xcite , this map was introduced by erds . since our approach in this paper is based on the conjectured generic property that all digits should become equally probable for ( products of ) high powers of digits , we have nothing new to say about @xmath62 ( although some results we have about @xmath64 can be interpreted in terms of @xmath65 ) . let us start by making a trivial but important remark . choose some base @xmath2 . looking back at the expansion given in , it is clear that we have the following _ trivial dichotomy _ :    1 .   _ either _ one of the digits @xmath66 is equal to zero , in which case @xmath67 ; 2 .   _ or else _ @xmath8 is equal to a product of digits in @xmath68 .    this trivial dichotomy , together with first observations of relevant but scarce numerical data , suggest us an obvious strategy . in order to answer sloane s question in the affirmative for base @xmath3 , it suffices to establish the following . [ basebpersist ] there exists a positive integer @xmath69 such that , for all @xmath70 , the base @xmath3 expansion of any product of @xmath71 digits greater than @xmath41 for base @xmath3 has at least one digit equal to zero . as soon as a product of digits becomes divisible by @xmath3 , a zero appears as its right - most digit when written in base @xmath3 , and therefore the next iteration of the sloane map yields @xmath72 as the result . this trivial remark allows us to concentrate only on those products that are not divisible by the base @xmath3 . the effect of that will be dramatic when @xmath58 . the following two conjectures are implied by conjecture [ basebpersist ] .    in the case of base @xmath43 , the only non - zero values assumed by the sloane map are powers of @xmath42 . in other words , we only need to investigate the orbit of @xmath41 under the doubling map @xmath73 . 1a[base3persist ] there exists a positive integer @xmath74 such that , for all @xmath75 , the base-@xmath44 expansion of @xmath76 has at least one digit equal to zero . such a statement is reminiscent of a conjecture by erds to the effect that there is always a 2 among the digits in base 3 of @xmath76 for all @xmath71 sufficiently large . this question has been recently adressed by lagarias in @xcite .    in the case of base @xmath58 , in principle all products of the form @xmath77 would need to be examined . but since the basis 4 divides @xmath77 as soon as @xmath78 , we only have to consider : ( a ) powers of 3 or ( b ) products of the form @xmath79 . in other words , we only need to consider the orbits of 1 and 2 under the tripling map @xmath80 . 1b[base4persist ] there exists a positive integer @xmath81 such that , for all @xmath82 , the base-@xmath83 expansions of @xmath84 and @xmath85 each have at least one digit equal to zero . there is considerable computational evidence in favor of these conjectures . proving these conjectures is certainly sufficient to solve the persistence problem for the corresponding bases . let us take the time to establish a weak estimate on @xmath86 , the minimum stability time of @xmath13 ( in base @xmath44 ) introduced in [ intro ] . this should be compared to a similar estimate proved by erds concerning the erds - sloane map . first , a very simple lemma . [ trivlem1 ] if @xmath87 , then at least one of the digits in the base-@xmath44 expansion of @xmath88 is not equal to @xmath42 . suppose the base-@xmath44 expansion of @xmath88 has exactly @xmath71 digits , all equal to @xmath42 . then @xmath89 , and we have a solution to catalan s equation @xmath90 with @xmath91 , which is impossible . see @xcite ) . the special case needed here ( with bases @xmath42 and @xmath44 , which are _ prime _ ) can be proved by elementary means see @xcite . ]    [ trivrem1 ] it follows from the proof that @xmath92 . let us now present our weak estimate . recall that if @xmath93 , then the number of digits in the base-@xmath3 expansion of @xmath94 is equal to @xmath95 .    for all @xmath96 , we have @xmath97 . let @xmath87 be given , and assume @xmath98 , otherwise there is nothing to prove . thus , suppose @xmath99 are such that    1 . @xmath100 for @xmath101 ; 2 . @xmath102 ; 3 . @xmath103 for @xmath104 ; 4 . @xmath27 is _ maximal _ with the above properties .    from these properties we see that the only possible values for @xmath105 are @xmath106 or @xmath107 . this implies that @xmath108 . hence it suffices to find a suitable bound for the number @xmath27 . let @xmath109 denote the number of digits in the base-@xmath44 expansion of @xmath110 , for @xmath111 . since @xmath100 , we deduce from lemma [ trivlem1 ] and remark [ trivrem1 ] that @xmath112 . but we also know that @xmath113 . therefore we have @xmath114 where @xmath115 . from this it follows that @xmath116 but we also have @xmath117 . hence , extracting the base-@xmath44 logarithm of all terms in the above inequality , we see that @xmath118 this shows that @xmath119 , as claimed . in this section , more precisely in  [ subsec : ergoz ] below , we introduce a simple dynamical system  generated by a single piecewise affine degree - one circle map  which will turn out to be very useful in the study of the sloane map for bases @xmath120 . but first let us present some simple arithmetical facts . looking at the base-@xmath44 expansions of the successive powers of two ( see figure [ fig : powersof2base3 ] ) , it is clear that for every @xmath121 the rightmost @xmath71 digits of these powers exhibit a periodic behavior . our goal here is to calculate the minimum period . this periodicity is more general : the last @xmath71 digits of the base-@xmath3 expansion of @xmath122 form a periodic sequence , for all @xmath123 . this fact is a simple consequence of modular arithmetic , and we call it _ tail periodicity_.    first , we need the following simple lemma . recall that , if @xmath2 is an integer , then the set of all residues @xmath124 modulo @xmath3 that are relatively prime with @xmath3 form a _ multiplicative group _ ( under multiplication modulo @xmath3 ) , called the _ group of units modulo @xmath3 _ , denoted @xmath125 . this group @xmath125 has order @xmath126 , where @xmath127 is euler s _ totient function_. an integer @xmath128 is said to be a _ primitive root modulo @xmath3 _ if the residue class of @xmath128 modulo @xmath3 has order @xmath126 in @xmath125 ( in particular , if such a primitive root exists , then the group @xmath125 must be cyclic ) . [ primroot1 ] let @xmath128 and @xmath3 be integers such that @xmath123 and @xmath128 is a primitive root modulo @xmath3 . then the residues @xmath129 , @xmath130 , form a periodic sequence with minimum period equal to @xmath126 .    by hypothesis , @xmath131 is a generator of the cyclic group @xmath125 . therefore , by euler s theorem , @xmath132 , and the elements @xmath133 exhaust the elements of @xmath125 . since @xmath134 for all @xmath1 , the sequence @xmath122 is indeed periodic , with minimum period @xmath126 .    as a consequence , we have the following . given integers @xmath121 and @xmath30 , let @xmath135 denote the residue of @xmath88 modulo @xmath84 .    [ primroot2 ] for each @xmath121 , the sequence @xmath135 , @xmath136 , is periodic with minimum period @xmath137 . it is well - known that @xmath42 is a primitive root modulo @xmath84 for all @xmath121 ( see @xcite ) . thus , the desired result follows at once from lemma [ primroot1 ] ( with @xmath138 and @xmath139 ) and the fact that @xmath140 . note that for each @xmath71 there are exactly @xmath137 strings of length @xmath71 in the symbols @xmath141 which do not end in a zero . call these strings _ allowable_. combining this counting , the above lemma and the fact that @xmath135 also never ends in a zero ( otherwise @xmath88 would be divisible by @xmath44 ) , we see that in any full ( minimum ) period of @xmath135 each allowable string appears exactly once . moreover , since there are exactly @xmath76 allowable strings of length @xmath71 that show only the digits @xmath41 and @xmath42 , the proportion of allowable strings in which at least one zero appears ( relative to the total number of allowable sequences of length @xmath71 ) tends to @xmath41 as @xmath142 . from this we deduce that the set @xmath143 has _ asymptotic density @xmath41 _ , in the sense of  [ sec : density ] below . this fact can be generalized to other bases ( but the counting argument used here becomes a bit more involved ) . rather than delve into such arithmetic methods , we shall use a completely different approach to prove a more general density result that holds true for all bases .      if , instead of looking at the final @xmath71 digits of the base-@xmath44 expansions of powers of two , we look at the first @xmath71 digits ( for @xmath121 fixed ) , the behavior of such _ heads _ is no longer periodic . it is rather _ almost periodic _ , as we will show ( see remark [ remalmost ] ) .     to      in fact , let us present a more general result . we consider two integers @xmath123 and consider the sequence @xmath122 , @xmath130 , written in base @xmath3 , say @xmath144_q\;=\;\sum_{i=1}^{k_n } d_{i , n}q^{k_n - i}\ , \ ] ] where @xmath145 and @xmath146 , and where the number of digits @xmath147 is given by @xmath148 . if we add a `` decimal point '' in front of the above expansion , we get the number @xmath149_q\;\in\;\delta_q \ , \ ] ] where @xmath150\subset \mathbb{r}$ ] . note that @xmath151 . now , the crucial observation is that the sequence @xmath152 just defined is precisely the orbit of @xmath153 under the piecewise - affine map @xmath154 given by @xmath155 this map is built out of two linear maps of the real line : one expanding ( multiplication by @xmath156 ) , the other contracting ( multiplication by @xmath157 ) ; see figure [ slope ] . through the identification of the endpoints of @xmath158 , the map @xmath159 becomes a ( piecewise affine ) homeomorphism of the circle . the reader can easily check that @xmath160 , for all @xmath30 . has at least one zero in its base 3 expansion for all @xmath161 , and as explained by tao in said blog for the collatz case . ] now we have the following result .    [ proptopconj ] the map @xmath154 is topologically conjugate to the rotation by @xmath162 . the conjugacy is lipschitz , in fact differentiable except at a single point . moreover , @xmath159 has an absolutely continuous invariant measure given by @xmath163    let us write @xmath164 in this proof . first note that @xmath165 where we have used that @xmath166 . hence , defining @xmath167\to \delta_q$ ] by @xmath168 we see that @xmath169 . in other words , @xmath170 maps the orbit of @xmath72 under the rotation @xmath171 onto the orbit of @xmath172 under @xmath173 . this suggests that @xmath170 is a conjugacy between @xmath174 and @xmath173 .    to see this , let @xmath175 $ ] , and note that there are two possibilities .    1 . we have @xmath176 : in this case , we have @xmath177 , and therefore @xmath178 2 . we have @xmath179 : in this case , @xmath180 , and therefore @xmath181    thus , it follows that @xmath182 . the map @xmath170 is a homeomorphism , as is clear from . in fact @xmath170 is an analytic diffeomorphism in the open interval @xmath183 . upon identification of the endpoints @xmath72 and @xmath41 , @xmath170 becomes a circle homeomorphism which is differentiable at all points except one , where a break in the derivative occurs . the absolutely continuous invariant measure @xmath184 for @xmath173 in @xmath158 can be obtained very simply as the push - forward of lebesgue measure @xmath185 in @xmath186 $ ] via @xmath170 . to wit , if @xmath187 is borel measurable , we have @xmath188 since from we have @xmath189 it follows that @xmath190 as was to be proved . [ remalmost ] note that we do _ not _ assume that @xmath162 is irrational in the above theorem . when @xmath191 is irrational , the orbit of any point under @xmath159 is almost periodic . hence we have the promised almost periodicity of tails . for a more general result concerning piecewise affine homeomorphisms of the circle , see @xcite .    [ largerp ] the above theorem also holds true when @xmath192 , provided we define @xmath159 appropriately in such cases . this can be done as follows . let @xmath121 be such that @xmath193 , and define @xmath194 then , just as before , @xmath154 is a piecewise affine degree - one circle map with rotation number @xmath195 , and this circle map preserves the same measure @xmath184 given in theorem [ proptopconj ] ( _ cf . _  [ sec : further ] ) .    in @xcite , boshernitzan gave an example of a piecewise affine homeomorphism of the unit circle @xmath196 ( seen as the interval @xmath186 $ ] with the endpoints identified ) which preserves the rationals and has only dense orbits . we remark that our @xmath159 s yield a _ plethora of such examples_. indeed , defining @xmath197\to \delta_q$ ] by @xmath198 , we see that each conjugated map @xmath199 is an example of the type devised by boshernitzan , provided its rotation number @xmath195 is irrational . let us now present two corollaries of theorem [ proptopconj ] . in order to state and prove the first corollary , we recall that the _ lower density _ of a subset @xmath200 , is given by @xmath201)\ .\ ] ] one defines the _ upper density _ @xmath202 of @xmath203 in similar fashion , replacing the @xmath204 by @xmath205 . if the upper and lower densities are equal , then we say that @xmath203 has _ asymptotic density _ recall also that a continuous self - map @xmath173 of a compact metric space @xmath207 is _ uniquely ergodic _ , _ i.e. _ has a unique invariant borel probability measure @xmath184 , if and only if its birkhoff time averages @xmath208 converge uniformly to the space average @xmath209 , for every @xmath210 ( see for instance @xcite ) . [ rankonedensity ] if @xmath123 are such that @xmath211 is irrational , then the set @xmath212 has asymptotic density equal to @xmath41 . let @xmath164 and @xmath184 be as in theorem [ proptopconj ] , and let @xmath213 be our special orbit as before . since @xmath173 is lipschitz - conjugate to an irrational rotation , it is uniquely ergodic . for each @xmath214 , let @xmath215 be the set of all points @xmath216 whose base-@xmath3 expansion has at least one digit @xmath72 among its first @xmath19 digits . then @xmath217 is a finite union of intervals , and its lebesgue measure is easily seen to be @xmath218\ ] ] thus , @xmath219 , and therefore @xmath220 , as @xmath221 . note that if @xmath222 for some @xmath1 with @xmath223 , then @xmath224 , _ i.e. _ @xmath225 . since @xmath217 is a _ finite _ union of intervals and @xmath184 is a regular borel measure , we can find a continuous function @xmath226 such that @xmath227 everywhere and @xmath228 in particular , for all @xmath229 we have @xmath230 using and the fact that @xmath173 is uniquely ergodic , we deduce that @xmath231 provided @xmath94 is sufficiently large . hence , combining and , we get @xmath232 for all sufficiently large @xmath94 . writing @xmath233 , we have just proved that @xmath234 since all but finitely many elements of @xmath235 belong to @xmath203 , it follows that @xmath236 as well , for every @xmath19 . letting @xmath221 , we deduce that @xmath237 , as desired . another consequence of theorem [ proptopconj ] is the following explicit formula for the @xmath19-th digit of @xmath122 written in base @xmath3 .    if @xmath238 denotes , as in , the @xmath19-th digit from left to right in the base @xmath3 expansion of @xmath122 , then @xmath239 where , as before , @xmath162 and @xmath147 is the number of digits in that expansion . again , we let @xmath164 . since @xmath238 is also the @xmath19-th digit of @xmath213 after the decimal point , we can certainly write @xmath240 but by theorem [ proptopconj ] , we have @xmath241 this immediately implies formula . although it may seem a bit surprising that we have such an explicit formula for the digits @xmath238 , the formula is in practice rather useless for large values of @xmath1 : roughly speaking , evaluating @xmath238 depends on knowing at least the first @xmath1 digits of @xmath162 after the decimal point .    at least for bases @xmath44 , @xmath83 , @xmath242 , and @xmath243 , we have strong computational evidence suggesting the validity of the following conjecture , which ( for base @xmath44 ) is certainly stronger than conjecture [ base3persist ] .    [ equaldigits ] if @xmath3 is not a power of @xmath128 , then for each @xmath244 , we have @xmath245 ( as before , @xmath148 is the number of digits of @xmath122 in base @xmath3 . )    for computational evidence and heuristic arguments supporting this conjecture , see  [ sec : compevid ] . if @xmath246 for some @xmath121 , then for every @xmath7 the base @xmath3 expansion of @xmath247 consists of the digit @xmath41 followed by @xmath1 zeros . this shows that the hypothesis that @xmath3 is not a power of @xmath128 is indeed necessary . we wish to generalize the results presented in  [ sec : erg1 ] to the cases when the base @xmath3 is greater than @xmath83 . the relevant dynamical system here is no longer the ( semi-)group given by a single piecewise affine degree - one circle map , but rather the ( semi-)group generated by several such maps . let @xmath248 be the list of all primes smaller than @xmath3 . if @xmath13 is such that @xmath249 , then we can certainly write @xmath250 where @xmath251 . we can treat the base-@xmath3 expansion of the right - hand side of pretty much in the same way as we treated the digits of @xmath122 in base @xmath3 in  [ subsec : ergoz ] . keeping the notation introduced in [ subsec : ergoz ] , let @xmath252\equiv \mathbb{s}^1 $ ] , and consider the piecewise affine homeomorphisms @xmath253 ( @xmath254 ) defined taking @xmath255 in . we denote by @xmath256 the group of all piecewise affine homeomorphisms of the interval @xmath257 . the group @xmath258 generated by @xmath259 is abelian . let @xmath167\to \delta_q$ ] be the homeomorphism constructed in the proof of theorem [ proptopconj ] . then for each @xmath260 we have @xmath261 , where @xmath262 and @xmath263 is the corresponding rotation . since any two circle rotations commute , we have @xmath264 , and therefore @xmath265 as well , so @xmath266 is abelian . this lemma tells us in particular that we have a well - defined surjective homomorphism @xmath267 given by @xmath268 however , this homomorphism is not necessarily one - to - one ( but see below ) . in any case , we see that @xmath269 acts on the circle in a special way as a group of piecewise - affine homeomorphisms of the circle . what is the relevance of this action to the study of the sloane map in base @xmath3 ? in order to answer this question , we proceed as in  [ subsec : ergoz ] . let @xmath270_q= \frac{1}{q}\in \delta_q$ ] . then for each @xmath271 we have @xmath272 where @xmath273 is the number of digits of the base-@xmath3 expansion of the numerator , given by @xmath274 thus , we see that the entire range of values assumed by the sloane map is contained in a single orbit of the action of the semi - group @xmath275 .      we shall need the following facts about _ free - abelian _ actions . let @xmath276 be a free abelian group of rank @xmath71 , with a fixed set of generators @xmath277 . each @xmath278 has a unique representation @xmath279 with @xmath280 for all @xmath281 . we write @xmath282 , the _ norm _ of @xmath283 . we denote by @xmath284 the semigroup consisting of all elements @xmath283 for which @xmath285 for all @xmath260 . for each @xmath93 , let @xmath286 , and let @xmath287 . now suppose that the group @xmath276 ( or the semigroup @xmath288 ) acts on a probability measure space @xmath289 as a group ( or semigroup ) of measure - preserving transformations ( m.p.t.s ) . in other words , if @xmath290 denotes the m.p.t . associated to @xmath278 , then @xmath291 for every @xmath283 ( where the star denotes push - forward of measures ) . we say that the @xmath276-action ( or @xmath288-action ) on @xmath289 is _ ergodic _ if the only measurable subsets @xmath292 that are invariant under @xmath276 ( or @xmath288 )  _ i.e. _ such that @xmath293 for all @xmath283  are either null - sets or full - measure sets . just as for rank - one measure - preserving actions , there is a multi - dimensional version of birkhoff s ergodic theorem , both in the case of free - abelian groups and free - abelian semigroups . we state the version for semigroups , which is the relevant one here . a[multiplebirkhoff ] if the @xmath288-action on @xmath289 is ergodic , then for every @xmath294 and for @xmath184-almost every @xmath295 we have @xmath296    a proof of this theorem can be found in ( * ? ? ? * ch .  2 ) . one can also define _ unique ergodicity _ by analogy with the rank - one case . a @xmath276-action ( or @xmath288-action ) on a compact metric space @xmath207 through continuous maps is _ uniquely ergodic _ if there exists a unique borel probability measure on @xmath207 which is @xmath276-invariant ( or @xmath288-invariant ) . as for rank - one actions , we have the following fact ( which , once again , we state only for semigroup actions ) . b[uniqueergod ] if a @xmath288-action by continuous maps on a compact metric space @xmath207 is uniquely ergodic then for every @xmath297 @xmath298 as @xmath299 , where @xmath184 is the unique @xmath288-invariant borel probability measure . the proof of the analogous statement for rank - one actions as given in , say , @xcite applies _ mutatis mutandis _ to the present case . the converse of theorem [ uniqueergod ] is also true , but will not be needed here . now we go back to our investigation of the sloane map . given a positive integer @xmath94 , let @xmath300 denote the `` cube '' @xmath301 , and let @xmath302 . the _ lower asymptotic density _ of a subset @xmath303 is defined to be @xmath304 the _ upper asymptotic density _ of @xmath203 , denoted @xmath202 , is similarly defined ( replacing @xmath204 by @xmath205 ) . we always have @xmath305 . when @xmath306 , this common value is called the _ asymptotic density _ of @xmath203 and it is denoted by @xmath307 . [ trivdensity ] if the base @xmath3 is not a prime number , then the set @xmath308 has asymptotic density equal to @xmath41 . since @xmath3 is not a prime , we can certainly write @xmath309 where each @xmath310 . now , if @xmath271 is such that @xmath311 for all @xmath260 , then @xmath3 divides @xmath312 , and therefore @xmath313 . this shows that @xmath314 . hence , for every @xmath94 sufficiently large we have @xmath315 and therefore @xmath316 this proves that @xmath237 . the set @xmath203 in proposition [ trivdensity ] has asymptotic density @xmath41 for trivial reasons : most @xmath27-tuples @xmath317 are such that @xmath311 for all @xmath260 . note that we did not exploit the abelian action introduced in [ sec : abelact ] . we will prove in  [ sec : secdensity ] below a more refined version of proposition [ trivdensity ] using the ergodic properties of such abelian actions . our second density result makes use of the group @xmath266 and its action on the circle . in fact , it will make use of certain subgroups of @xmath266 , which turn out to be _ free abelian_.    we will need the following lemma . given @xmath318 , let us denote the subgroup of @xmath266 generated by @xmath319 by @xmath320 . recall that @xmath262 is the rotation number of @xmath321    [ freeuniqergod ] if the numbers @xmath322 are rationally independent , then @xmath320 is a free abelian group of rank @xmath71 . moreover , its action on the circle @xmath323 is uniquely ergodic . suppose @xmath324 is such that @xmath325 from this and the fact that each @xmath321 is conjugate to @xmath326 by the same conjugating map , we have @xmath327 but then @xmath328 . in other words , there exists @xmath329 such that @xmath330 . the hypothesis of rational independence implies that @xmath331 . hence there are no non - trivial relations in @xmath320 . this shows the group is free abelian as stated . moreover , at least one of the @xmath332 s must be irrational . say @xmath333 is irrational ; then @xmath334 , being conjugate to an irrational rotation , is uniquely ergodic . therefore , _ a fortiori _ , the action of @xmath320 on the circle is uniquely ergodic . we are now in a position to state and prove our second density result . this result refines proposition [ trivdensity ] , and in particular covers the case when the base @xmath3 is prime . [ rankkdensity ] let @xmath318 be chosen so that the numbers @xmath322 are rationally independent . then for every divisor @xmath335 of @xmath3 , the set @xmath336 has asymptotic density equal to @xmath41 . things have been set up so that the same argument used in the proof of corollary [ rankonedensity ] can be applied , _ mutatis mutandis_. by lemma [ freeuniqergod ] , the action of the group @xmath337 on the circle @xmath323 is uniquely ergodic ; the unique invariant measure is the measure @xmath184 constructed in theorem [ proptopconj ] . let us fix the divisor @xmath335 of the base @xmath3 . we are interested in a particular orbit of the semigroup @xmath338 , namely that of the point @xmath339 . for each @xmath340 , let us write @xmath341 . also , denote by @xmath342 the number of digits in the base-@xmath3 expansion of the number @xmath343 . then , just as in , we have @xmath344 as in the proof of corollary [ rankonedensity ] , for each @xmath214 , let @xmath215 be the set of all points @xmath216 whose base-@xmath3 expansion has at least one digit @xmath72 among its first @xmath19 digits . as we saw there , @xmath220 as @xmath221 . let @xmath345 be as in that proof also ; thus , each @xmath346 is continuous and @xmath347 , and @xmath348 the point now  as is clear from  is that @xmath349 in other words , @xmath350 if and only if @xmath351 for some @xmath352 . by analogy with what we did in the proof of corollary [ rankonedensity ] , for each @xmath214 we define @xmath353 . note that all but finitely many elements of @xmath235 belong to @xmath203 . hence , in order to prove that @xmath354 , it suffices to show that @xmath355 as @xmath356 . since the action of @xmath288 on the circle is uniquely ergodic , combining theorem [ uniqueergod ] with we deduce that , for each fixed @xmath19 and all sufficiently large @xmath94 , @xmath357 since @xmath358 , it follows that @xmath359 for all sufficiently large @xmath94 . letting @xmath360 , this shows that @xmath361 for all @xmath19 , and therefore @xmath355 as @xmath356 , as required . since in this paper we are primarily interested in the sloane map , in all of the above we have focused on products of prime numbers smaller than the base @xmath3 . however , most of what we have done goes through when some or all of such primes are greater than @xmath3 . recall from remark [ largerp ] that if @xmath192 we can still define the circle maps @xmath159 , and these still commute with each other because they all share a common absolutely continuous invariant measure . thus , suppose that @xmath362 is some non - empty finite set of primes , and that not all the prime divisors of @xmath3 are in @xmath362 . then the set @xmath363 is rationally independent . therefore the maps @xmath159 for @xmath364 generate a free - abelian group @xmath365 and the action of @xmath365 on the circle @xmath158 is uniquely ergodic ( as in lemma [ freeuniqergod ] , and the proof is the same ) . moreover , the density result given in theorem [ rankkdensity ] also holds true for the list of primes in @xmath362 . in particular , the vast majority of products of the form @xmath366 ( with @xmath367 ) have at least one digit zero in their base-@xmath3 expansions . in  [ sec : compevid ] we will make a much stronger assertion , in the form of a conjecture , to the effect that the digits in the base-@xmath3 expansions of such products become asymptotically equidistributed as @xmath368 . to ease many statements to be made in rest of the paper , we will call _ sloane s conjecture _ the positive answer to the question about multiplicative persistence raised by sloane in 1973 . our goal is to place sloane s conjecture as a consequence of a much more general conjecture concerning long products of primes ( chosen from a given finite set ) . as in  [ sec : further ] , let us be given a base @xmath2 and a finite set of primes @xmath362 with the property that not all prime divisors of @xmath3 belong to @xmath362 . we call such an @xmath362 an _ allowable set of primes for @xmath3_. suppose we play the following game : starting with any given positive integer @xmath369 , we randomly select a sequence @xmath370 of primes in @xmath362 and we use them to generate the sequence of products @xmath371 with @xmath13 . then it turns out that , regardless of the initial seed @xmath369 and of the sequence of primes selected , the frequency of each digit @xmath372 in @xmath373 always seems to approach @xmath374 as @xmath375 . more seems to be true , and we formulate the conjecture that was revealed by our numerical computations in the following elementary way . 3[conj : convvtoequi0 ] ( elementary formulation).given an integer @xmath2 , an allowable set of primes @xmath362 for @xmath3 , and a positive integer @xmath369 , consider any of the possible sequences of products defined by setting @xmath376 and then , for each @xmath377 , @xmath378 where each @xmath379 is a element of @xmath362 . then the digits @xmath380 are asymptotically equidistributed ( their numbers tend to be in equal proportions ) when @xmath375 in the base-@xmath3 representations of these successive @xmath373 s . furthermore , the same holds true for blocks of consecutive digits of any length , _ i.e. , _ the asymptotic proportion of each block of digits of length @xmath381 is given by @xmath382 , the reciprocal of the number of distinct blocs of length @xmath383 in base @xmath3 . we remark that when the restriction of allowability is removed ( but @xmath362 remains finite ) it seems that the non - zero digits remain well distributed while the zeros may be much more abundant , which is more than we need in order to get the sloane conjecture as a corollary . let us give a more precise and more general statement of the above conjecture , expressing it in the language of _ the patterns arising by the multiplication game described above strongly suggest analogy with the dynamics of some simple _ cellular automata _ ( abbreviated @xmath384 ) _ i.e. , _ automorphisms of a shift space that commute with the shift @xcite . successsive multiplications by any @xmath156 ( or by any sequence of such @xmath128 s ) do not yield a @xmath384 , because of the _ carryover effect _ , but give rise to a close enough object , which we define as follows . [ quasicellular ] let @xmath2 and let @xmath362 be an allowable set of primes for @xmath3 . a multiplication automaton ( or @xmath385 ) , with alphabet @xmath386 consists of a finite sequence of primes @xmath387 ( with @xmath7 ) called the multipliers , and two maps , the configuration map @xmath388 and the carryover map @xmath389 satisfying the following rules for all @xmath7 and all @xmath390 :    1 . @xmath391 ; 2 . @xmath392 .    here , we assume that the initial row @xmath393 of the configuration map is given , as well as the column @xmath394 of initial carryovers . we also assume that @xmath395 for all but finitely many values of @xmath260 , and we call the number @xmath396 the _ seed _ of the @xmath385 . note that each row of an automaton has only finitely many non - zero elements ; in other words , for each @xmath13 there exists a smallest @xmath397 such that @xmath398 for all @xmath399 . if the initial seed is non - zero , then @xmath400 as @xmath375 .    despite appearances from the above recursive formulas , it turns out that the values of the configuration map of a @xmath385 are determined _ purely locally _ , as the following proposition shows . this further reinforces the similarity of @xmath385 s with @xmath384 s . [ prop : locality ] in every @xmath385 , the configuration value @xmath401 depends only on the three values @xmath402 and on the multiplier @xmath403 . we refer to formulas ( i ) and ( ii ) in definition [ quasicellular ] . from ( ii ) with @xmath404 replacing @xmath260 we see that @xmath405 but formula ( i ) with @xmath404 replacing @xmath260 gives us @xmath406 combining and with ( i ) we deduce that @xmath407 \ , , \ ] ] which is the desired result . the way the @xmath385 operates purely locally as told by proposition [ prop : locality ] is illustrated in part ( a ) of figure [ fig : qauto ] where in each group of positions , the one framed in black is the value of @xmath401 . it is worth pointing out that @xmath385 s can be interpreted as examples of _ error diffusion _ where the input is the previous line , the modified input is the product of the input by @xmath403 added to the carry over , and the new error is the next carry over ( see @xcite and references therein ) . incidentally , part ( b ) of the same figure gives the allowed configuration of another automaton , namely the one that describes the evolution along columns ( a simple model of discrete epitaxy ) . the configurations in part ( c ) of that figure are the forbidden ones resulting in a solid black square when nothing can be computed . the simplest example of a multiplication automaton relevant to sloane s conjecture is obtained by taking @xmath43 and @xmath408 , choosing @xmath409 as the seed , and setting @xmath410 for all @xmath1 as the initial sequence of carryovers . note that in this case @xmath411 for all @xmath1 . the configuration map for this automaton yields the list of all powers of @xmath42 written in base @xmath44 . this list corresponds to the orbit of @xmath412_3 $ ] under the map @xmath413 introduced in  [ subsec : ergoz ] . in the notation introduced above , we have @xmath414 for all @xmath1 . figure 2 exhibits the first 45 rows of this @xmath385 , _ i.e. _ the first 45 powers of 2 ( out of the about 8000 that we have computed ) . some patterns seen there are reminiscent of those appearing in the simplest @xmath384 s .      another pair of examples is obtained by taking @xmath58 and @xmath415 , setting @xmath410 for all @xmath1 as before , and choosing either @xmath409 or @xmath416 as the seed . the resulting pair of @xmath385 yields the two sequences @xmath417 and @xmath418 written in base @xmath83 , which are the relevant ones for sloane s conjecture in that base . these sequences correspond to the orbits of @xmath419_4 $ ] and @xmath420_4 $ ] under successive iterations of the map @xmath421 introduced in  [ subsec : ergoz ] . the 23 first rows of these automata are presented side by side in figure 3 . a @xmath385 as given in definition [ quasicellular ] can be used as a statistical model for the discrete - time evolution of a mixture of @xmath3 species ( labeled balls , say , or distinct molecules ) with positions at time @xmath1 labeled by the integers @xmath422 . one is then interested in the asymptotic behavior of the mixture as characterized by the proportions of species populations as @xmath375 . this analogy with statistical mechanics and the observed patterns in the two examples above ( see figures 2 and 3 ) , as well as in several other examples we have investigated , suggest the following conjecture . is identically zero . ] 3[conj : convvtoequi ] ( general formulation . ) the configuration map of every multiplication automaton with non - zero seed converges to an equilibrium in the following sense . for each @xmath423 and each @xmath424-block @xmath425 , we have @xmath426    in other words , the populations of the @xmath3 species in the mixture become asymptotically perfectly balanced as time evolves . figure 4 shows clearly that the proportion of zeros in @xmath88 ( _ i.e. _ , in row @xmath1 of the automaton of example 1 ) goes to one third of the digits when @xmath1 becomes large enough . of course there are fluctuations in the proportions but the sizes of these fluctuations go to zero , as one would expect in a model of statistical physics . similar data have been obtained with bases @xmath83 , @xmath242 , and @xmath243 and a variety of allowable sets @xmath362 of prime multipliers . in many cases , we also checked that the expected asymptotic statistics of _ blocks _ behave according to the conjecture . besides potential applications to random number generators , conjecture [ conj : convvtoequi ] , if proved true , would yield several interesting corollaries , such as sloane s conjecture for all bases ( as we have already pointed out ) , and other conjectures by erds , furstenberg , and lagarias that are reported in @xcite .    to      note that in definition [ quasicellular ] we have assumed that the allowable set of primes @xmath362 from which the multipliers are chosen is _ finite_. if this condition is removed , conjecture [ conj : convvtoequi ] becomes false , as the following example shows . we work in base @xmath45 here , but similar examples can be given in other bases . recall from elementary number theory that a _ repunit _ is a positive integer whose ( base 10 ) expansion consists of a string of ones , e.g. @xmath427 , etc . if a repunit @xmath428 has @xmath1 digits , then of course @xmath429 let @xmath71 be any integer greater than @xmath41 and consider the infinite sequence of repunits @xmath430 given by @xmath431 then the identity @xmath432 shows that @xmath433 divides @xmath434 , for each @xmath7 . moreover , since @xmath435 , we see that @xmath434 has at least one prime factor which does not divide @xmath433 . hence there exist a sequence of primes @xmath370 ( not necessarily distinct , but ranging over infinitely many values ) and a sequence of natural numbers @xmath436 such that @xmath437 for all @xmath1 . but this means that each @xmath433 appears as a row of the ( generalized ) @xmath385 with base @xmath45 , seed @xmath409 and with @xmath362 being the set of all primes appearing as factors of some @xmath433 , which is infinite as we have shown . in particular , the line - by - line digits for this automaton can not be asymptotically equidistributed . therefore conjecture [ conj : convvtoequi ] can fail to hold when @xmath362 is infinite . written in the decimal system , say , contains infinitely many primes . ] let us now give an chain of heuristic arguments lending further support in favor of conjecture [ conj : convvtoequi ] . the heuristics will be formulated for the most part in the context of example 1 above ( @xmath43 and @xmath408 ) . the key idea is to model the deterministic behaviour of an @xmath385 by a stochastic process . the irregularities observed in the rows of an @xmath385 are due to the carryover effect , and these will be interpreted as random . this will be our standing assumption . stated informally : _ everything is as if all would be random with appropriate distributions_.      as pointed out already , we restrict our analysis almost entirely to the @xmath385 of example 1 . we make the further restriction of looking only at the evolution of the populations of the @xmath44 species given by the single digits @xmath141 . we ignore , of course , all zeros in positions to the left of the last non - zero digit in each row . although our multiplication automaton is a _ object , we will model its evolution  more precisely the row - by - row evolution ( or growth ) of the population frequencies ( or proportions ) of the digits @xmath141  by a _ stochastic process_. once more , we leave to the reader to see how what we propose here can be extended to the general situation of the conjecture .    in the example at hand , the passage from row @xmath1 to row @xmath438 of the multiplication automaton involves only multiplication by @xmath42 . the only possible values of the carryovers are @xmath72 and @xmath41 ; in the general case of multiplication of @xmath439 in base @xmath3 , finding the maximal carryover is indeed the first computation to be done . thus , for each @xmath440 and each @xmath30 we have either @xmath441 if there is no carryover at the column position @xmath260 , or @xmath442 if there is a carryover at that position . this means that in each position @xmath260 , the possible transitions of digits from one row to the next are @xmath443 , @xmath444 and @xmath445 without carryover , and @xmath446 , @xmath447 and @xmath448 with carryover . these spell out the _ incidence matrix _ @xmath449 this suggests that the evolution of the proportions @xmath450 ( with @xmath451 ) of the digits @xmath141 from row to row of the automaton can be modeled by a simple ( homogeneous ) markov chain . we refer to the column vectors @xmath452 as _ population vectors_. our standing assumption in this context is that _ the allowable transitions from each given state have equal probabilities_. hence this markov chain has as its _ transition matrix _ the doubly stochastic matrix @xmath453 . the situation is not quite so simple , however , because this proposed scheme fails to take into account the fact that the _ total population _ @xmath147 occasionally increases as we go from row to row ( @xmath400 as @xmath375 ) depending on the leading digit , the multiplication factor , and the carryover inherited there.this growth of population size is our only hope to see the asymptotic equidistribution of digits stated above . indeed , we know from  [ sec : tails ] that the tails of size @xmath454 , namely the blocks @xmath455 ( with @xmath136 ) form a _ periodic sequence_. the same is true of the sequence of @xmath424-blocks @xmath456 for every fixed @xmath457 . hence there is no hope to observe equidistribution of digits along blocks of fixed sizes . this compels us to change the model slightly , to accommodate the eventual increase in population size . only two things can happen when we go from row @xmath1 to row @xmath438 : either @xmath458 , _ i.e. _ , the total population size stays the same , or @xmath459 , _ i.e. _ , the total population size grows by @xmath41 . in the first case , the new proportions at time @xmath438 can be obtained from the old proportions at time @xmath1 by multiplication with the matrix @xmath460 . in the second case , we can think that the left - most digit at row @xmath1 , namely @xmath461 gives rise not to one , but two new digits at row @xmath438 , namely @xmath462 and @xmath463 . of these two , the last one is always equal to @xmath41 . since we either have @xmath464 or @xmath465 , we now either have a non - zero probability of a transition @xmath466 , or the probability of a transition @xmath445 is now slighty greater than @xmath467 . in other words , the new population vector @xmath468 is obtained from the old population vector @xmath469 by multiplication with one of the two stochastic matrices @xmath470 where @xmath471 accounts for the increase of the population of @xmath41 s by one . summarizing , the evolution of population proportions in each row of the multiplication automaton in example 1 can be modelled by an _ inhomogeneous markov chain _ ( imc ) , given by a sequence of stochastic transition matrices @xmath472 and ( column ) probability vectors @xmath473 such that @xmath474 . here , each transition matrix @xmath475 . there is a well - developed theory of imc s ; see for instance @xcite , or @xcite . as it turns out , convergence to equilibrium as expressed in conjecture [ conj : convvtoequi ] is tantamount , in the present example , to _ strong ergodicity _ of the imc . let us explain this point . for each pair of integers @xmath476 , let @xmath477 . following , @xcite , we say that the imc is _ strongly ergodic _ if there exists a row probability vector @xmath478 such that @xmath479 where the supremum is over all column probability vectors @xmath480 , and where @xmath481 is the so - called _ distance in variation _ , defined by @xmath482 whenever @xmath483 and @xmath484 are ( row ) probability vectors . recall ( see @xcite , or any standard text on markov chains ) that a _ stationary distribution _ for a stochastic matrix @xmath485 is a column probability vector @xmath486 such that @xmath487 . we have the following sufficient condition for strong ergodicity , as stated in ( * ? ? ? 6.8.5 ) ( _ cf . v.4.5 , p. 170 ) ) . c if each transition matrix @xmath488 has at least one stationary distribution and if there exists a stochastic matrix @xmath489 such that @xmath490 as @xmath375 , then the imc is strongly ergodic . is the _ max - norm _ : if @xmath491 , then @xmath492 . ]    the hypotheses in this theorem are met in our example . indeed , here we clearly see from that @xmath493 and @xmath494 as @xmath495 ; hence we can take @xmath496 . moreover , an easy computation shows that the probability vectors @xmath497 are stationary distributions for @xmath498 and @xmath499 , respectively . the heuristic argument just presented supports the validity of conjecture [ conj : convvtoequi ] in the case of example 1 , as long as one accepts the standing assumption stated in the beginning of this section . the formalism of imc s can be similarly used to treat @xmath385 s with any other bases @xmath3 and allowable sets of multipliers @xmath362 . the analysis of the two @xmath385 s in example 2 is completely analogous , because in those two cases @xmath415 is still a unitary set . things become a bit different when @xmath362 has two or more elements . for example , take @xmath500 and @xmath501 . then instead of a single ( unperturbed ) stochastic matrix @xmath460 we now have a pair of such matrices , one for each multiplier : @xmath502 in each of these two matrices , the number of non - zero entries in each row equals the number of possible carryovers ( upon multiplication by the corresponding prime ) . as we have seen in the previous subsection , in the context of example 1 , each digit @xmath503 in a given position on row @xmath1 of our @xmath385 gives rise to a new digit @xmath504 immediately below it on row @xmath438 , where either @xmath505 or @xmath506 , depending on whether the carryover at that position is @xmath72 or @xmath41 , respectively . we have stated that , if the carryovers in row @xmath507 are assumed to be randomly placed , then the transition probabilities for the digit transitions @xmath508 are equal to @xmath467 ( hence the stochastic matrix @xmath460 introduced earlier ) . strictly speaking , this is not correct . but as we will see below , the statement is close to being true provided the digits on row @xmath1 are already approximately uniformly distributed in the following ( finite ) sense . we say that a @xmath71-block @xmath509 _ occurs _ in an @xmath94-block @xmath510 if there exists @xmath511 such that @xmath512 . an @xmath94-block @xmath513 is said to be @xmath71-balanced ( for a given @xmath71 with @xmath514 ) if for each @xmath71-block @xmath515 the total number of ocurrences of @xmath516 in @xmath517 divided by the total number of @xmath71-blocks occurring in @xmath517 is equal to @xmath518.-blocks is @xmath84 . ] the rough idea is that if row @xmath1 of our @xmath385 is @xmath71-balanced and we count , for a given allowable transition @xmath508 , how many times this transition occurs when we go from row @xmath1 to row @xmath438 and divide that number by the total number of observed transitions which start with the digit @xmath216 , then this ratio  which we call an _ empirical transition probability _  is approximately equal to @xmath467 , with an error which is exponentially small in @xmath71 . we call this phenomenon _ block protection_. the result can be formulated as follows .    [ protectblock ] if the @xmath1-th row @xmath519 is @xmath71-balanced for some @xmath121 , then the empirical probability @xmath520 of each transition @xmath508 ( computed from the digit transitions @xmath521 ) satisfies @xmath522 .    the proof is deferred to appendix [ appendix ] .    instead of looking at transitions @xmath508 between digits , we may consider more generally transitions @xmath523 between blocks @xmath524 of a fixed length @xmath424 . here @xmath525 lies on row @xmath1 of our @xmath385 while @xmath526 lies on row @xmath438 immediately below @xmath207 . for each given @xmath207 there are two possible values for @xmath527 ( depending on the carryover @xmath528 ) . one can talk about block protection of such _ blocks _ in the same way we talked about block protection of _ digits_. an exact analogue of proposition [ protectblock ] holds true if we simply replace @xmath216 by @xmath207 and @xmath529 by @xmath527 , and the proof is similar . the end result is that _ if row @xmath1 of our @xmath385 is @xmath71-balanced _ , then the empirical transition probabilities @xmath530 differ from @xmath467 by an error smaller than @xmath531 ( where @xmath71 is the length of the protecting blocks ) . to be more specific , let @xmath532 be the stochastic matrix that gives the actual transitions of @xmath424-blocks from row @xmath1 to row @xmath438 , and _ assume _ that row @xmath1 is @xmath71-balanced . for @xmath533 there are @xmath534 blocks . if all allowed transitions were equally likely , we would get the @xmath535 stochastic matrix @xmath536 the actual stochastic matrices @xmath537 differ from the above matrix only in its non - zero entries , and by less than @xmath531 , _ provided row @xmath1 is @xmath71-balanced_. more generally , for arbitrary @xmath424 , if all allowed transitions @xmath523 were equally likely , the associated stochastic matrix would be @xmath538_{1\leq i , j\leq 3^\ell } \ , \ ] ] where @xmath539 stands for kronecker s delta . once again , the actual stochastic matrices @xmath540 differ from the matrix in only in its non - zero entries , and by less than @xmath531 _ if row @xmath1 is @xmath71-balanced_.      now we come to the last part of the heuristics . almost all of what was done in parts i and ii of the heuristics can be rigorously proved . this is not the case , so far , with the arguments in the present section . the discussion will be more informal . the imc formalism as described in  [ imcsection ] can only work if the stochastic matrices @xmath540 introduced in [ sub : heuristic2 ] become asymptotically closer and closer to the corresponding matrices @xmath541 as @xmath495 , for any given @xmath424 . thus , we need @xmath542 for any block - length @xmath424 . this is indeed corroborated by our computational evidence . proposition [ protectblock ] gives us a much weaker , conditional result : _ if _ the @xmath1-th row @xmath373 of our @xmath385 is @xmath71-balanced , _ then _ @xmath543 . it is of course too much to expect that a row of our @xmath385 will be perfectly @xmath71-balanced . but it is not difficult to generalize proposition [ protectblock ] so that the expression `` @xmath71-balanced '' is replaced by a suitable concept of `` almost @xmath71-balanced '' ( which can be stated in an appropriate quantitative way ) , so that the estimate in the conclusion is almost the same , with @xmath531 replaced by @xmath544 , say . let us agree to say that the matrix @xmath540 is _ @xmath71-reasonable _ if @xmath545 . the stability of the arguments presented so far then tell us that if row @xmath1 of our @xmath385 is almost @xmath71-balanced , then the stochastic matrices @xmath546 , @xmath547 remain @xmath71-reasonable for a certain number @xmath548 of steps . thus , in order to justify the use of the imc formalism and get a proof of conjecture [ conj : convvtoequi ] ( at least for the case of the @xmath385 of example 1 ) , we need two things :    1 . to guarantee the possibility of a _ good start _ , _ i.e. , _ a row @xmath1 of our automaton which is almost @xmath71-balanced , for as large a value of @xmath71 as possible . 2 .   to make sure that throughout the steps @xmath549 the longer blocks in @xmath550 ( _ i.e. , _ those whose evolutions are not protected by the distributions of even longer blocks ) are not progressively getting too unevenly distributed , with consequences that would then cascade down to shorter and shorter blocks .    at least at a rough level , point ( 1 ) is not difficult to achieve thanks to our results in  [ sec : erg1 ] . indeed , by what we have seen in that section , adding a decimal point in front of @xmath373 ( the @xmath1-row of our @xmath385 ) , we get the orbit @xmath551 of @xmath270_3 $ ] under the circle map @xmath413 . since all orbits of @xmath413 are dense , given any @xmath71-balanced string @xmath552 ( with @xmath553 ) , there exists @xmath7 such that @xmath554 has the string @xmath527 as its @xmath27-prefix , or head , which we call _ good_. for each such @xmath1 with a good head , the corresponding @xmath373 has a suffix , or tail , of a certain length @xmath457 . note that we can not choose @xmath457 _ a - priori _ : if one uses the density of orbits of @xmath413 , one is forced to accept the value of @xmath457 imposed by the choice of the good head @xmath527 . now , as we iterate further and consider the successive rows @xmath555 of the automaton , the tail of length @xmath457 generates a periodic sequence of tails lying beneath it ( all with the same length @xmath457 ) , whereas the heads increase in size . clearly , the constant character of @xmath457 as @xmath19 increases makes the tails irrelevant in the computation of asymptotic proportions of symbols or other blocks . the problem then becomes point ( 2 ) above for the heads . this point is experimentally verified but is mathematically beyond our reach at this writing , contrary to point ( 1 ) as formulated above . as promised , in this section we prove proposition [ protectblock ] . the proof will require the two lemmas presented below . let us fix @xmath121 . it is an easy consequence of proposition [ prop : locality ] that , for each @xmath556 and each @xmath7 , the element @xmath557 of our @xmath385 is completely determined by the following data :    1 . the element @xmath401 lying immediately above @xmath557 ; 2 .   the @xmath71-block @xmath558 lying to the right of @xmath401 ; 3 . the carryover @xmath559 , _ i.e. _ the carryover immediately to the right of @xmath560.t    in other words , there exists a function @xmath561 such that @xmath562 in fact , the function @xmath563 can be explicitly computed . define the _ value _ of a @xmath71-block @xmath564 by @xmath565 note that every block @xmath516 is uniquely determined by its value . we now have the following lemma .    for each digit @xmath566 , each @xmath71-block @xmath567 and each carryover @xmath568 , we have @xmath569    the concatenated block @xmath570 has value @xmath571 . in order to compute @xmath572 , we multiply this value by @xmath42 and add the carryover @xmath573 , getting the number @xmath574 . this number is written in base @xmath44 and the resulting block is placed beneath @xmath575 . the digit @xmath529 ( immediately below @xmath216 ) is precisely the @xmath71-th digit of @xmath576 from right to left , _ i.e. , _ @xmath577 . hence we have @xmath578 and this proves . note that the last term in the right - hand side of is equal to either @xmath72 or @xmath41 . now , it turns out that @xmath563 is `` almost '' independent of the variable @xmath568 . roughly speaking , the only way the carryover @xmath573 to the right of the @xmath71-block @xmath516 can influence the value of the digit @xmath529 immediately below @xmath216 ( on the left of @xmath516 ) is if @xmath516 happens to be the block @xmath579 ( @xmath71 times ) . every other block will contain in some position a @xmath72 or a @xmath42 ; upon multiplication by @xmath42 these yield @xmath72 and @xmath41 , respectively , and any carryover effect coming from the right of that position will not go through to the left of it . this is part ( i ) of the following lemma . [ phiproperties ] the function @xmath563 has the following properties .    1 .   if @xmath566 and @xmath580 , then @xmath581 if and only if @xmath582 ; 2 . if @xmath566 and @xmath504 , then @xmath583    first , let @xmath516 be a @xmath71-block such that @xmath581 . then from we see that @xmath584 hence we must have simultaneously @xmath585 from these two inequalities we deduce that @xmath586 , and therefore @xmath587 . this means that @xmath582 . conversely , if @xmath582 then by a simple computation we see that holds true . this proves ( i ) .    to prove ( ii ) , there are two cases to consider : @xmath588 and @xmath589 . in either case , if @xmath590 then by part ( i ) we have @xmath591 if @xmath588 , then both sides in are equal to @xmath72 , and we get @xmath592 . the number of blocks @xmath516 satisfying these inequalities is precisely @xmath593 . if @xmath589 , then both sides in are equal to @xmath41 , and this time we deduce that @xmath594 . the number of blocks @xmath516 satisfying these last inequalities is also @xmath593 . this finishes the proof .    to      consider an allowable transition of digits @xmath508 from row @xmath1 to row @xmath438 of our @xmath385 . we are assuming that row @xmath1 is @xmath71-balanced . in order to compute the empirical probability @xmath520 of such transition , we need to count how many times this transition happens and divide it by the total number of transitions which start with @xmath216 on row @xmath1 . for this purpose , first we count how many blocks @xmath590 are such that @xmath595 . the answer is given by lemma [ phiproperties](ii ) : there are @xmath593 such blocks . since row @xmath1 of our @xmath385 is @xmath71-balanced , the proportion of such blocks in that row is therefore the quotient @xmath596 . this already tells us that @xmath597 we still have to account for the occurences of the block @xmath582 . let @xmath598 be such that @xmath599 . each occurrence of @xmath582 on row @xmath1 for which the carryover immediately to the right of @xmath516 equals @xmath573 contributes to the desired empirical probability . since the proportion of such occurrences is at most @xmath600 , this shows that @xmath601 combining with we get the inequality in the statement .    to    to the authors are grateful to george hentchel for bringing this problem to the attention of c.t . , and for sharing insights on how asymptotic equidistribution of digits would solve sloane s problem in base @xmath44 .
we investigate the so - called persistence problem of sloane , exploiting connections with the dynamics of circle maps and the ergodic theory of @xmath0 actions . 
 we also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would , in particular , solve the persistence problem . 
 the heuristics that we propose to complement our numerical studies can be thought in terms of a simple model in statistical mechanics .