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let @xmath8 and @xmath1 be the space of @xmath2-frames in @xmath3 ( i.e. the space of @xmath2-tuples of linearly independent vectors in @xmath3 ) , @xmath9 . the group @xmath10 acts on this space as follows : @xmath11 the action is transitive for @xmath12 . let @xmath5 be a lattice in @xmath10 ; that is , a discrete subgroup in @xmath10 such that the factor space @xmath13 has finite volume ( e.g. @xmath14 ) . the main result of this paper concerns distribution of @xmath5-orbits in @xmath1 .    when @xmath15 , every orbit of @xmath5 is discrete the situation becomes much more interesting for @xmath12 . let us recall known results :    * ( dani , raghavan @xcite ) * [ th_dr ] let @xmath14 , and @xmath16 be an @xmath2-frame in @xmath3 , @xmath4 . then the orbit @xmath17 is dense in @xmath1 iff the space spanned by @xmath18 contains no nonzero rational vectors .    * ( veech @xcite ) * [ th_ve ] if @xmath5 is a cocompact lattice in @xmath10 , then every orbit of @xmath5 in @xmath1 , @xmath4 , is dense . theorems [ th_dr ] and [ th_ve ] provide examples of dense @xmath5-orbits in @xmath1 . here we show that dense @xmath5-orbits are uniformly distributed with respect to an explicitly described measure on @xmath1 . this measure is @xmath19 , where @xmath20 is the lebesgue measure on @xmath21 , and @xmath22 is the @xmath2-dimensional volume of the frame @xmath23 . note that the measure @xmath20 is @xmath10-invariant , and it is unique up to a constant . however , orbits of @xmath5 are equidistributed with respect to the measure @xmath19 , which is not @xmath10-invariant . this phenomenon was already observed by ledrappier @xcite . define a norm on @xmath24 by @xmath25 for @xmath26 , @xmath27 , @xmath28 , put @xmath29 we determine asymptotic behavior of @xmath30 as @xmath31 . this result gives a quantitative strengthening of theorems [ th_dr ] and [ th_ve ] , and it can be interpreted as uniform distribution of dense orbits of @xmath5 in @xmath1 . [ th_frames00 ] let @xmath5 be a lattice in @xmath32 . let @xmath33 be an @xmath2-frame in @xmath3 such that @xmath34 is dense in @xmath35 . let @xmath36 be a relatively compact borel subset of @xmath35 such that @xmath37 . then @xmath38 where @xmath39 is a constant ( which is computed in ( [ eq_anl ] ) below ) , and @xmath40 is a @xmath10-invariant measure on @xmath13 ( which is defined in ( [ eq_mubar ] ) below ) . the term @xmath41 in ( [ eq_f_main00 ] ) comes from the asymptotics of the volume of the set @xmath42 in the stabilizer @xmath43 of @xmath44 with respect to the measure on @xmath43 which is determined by the choice of the haar measures on @xmath10 and @xmath45 ( see section [ sec_ttt ] ) .    for @xmath46 and @xmath47 , this theorem was proved by ledrappier @xcite for general @xmath5 and by nogueira @xcite for @xmath48 and @xmath49-norm using different methods . combining theorems [ th_dr ] and [ th_frames00 ] , we get :    [ th_frames ] let @xmath14 . let @xmath50 be an @xmath2-frame in @xmath3 such that the space @xmath51 contains no nonzero rational vectors . let @xmath36 be a relatively compact borel subset of @xmath35 such that @xmath37 . then @xmath52 where @xmath53 is a constant ( which is computed in ( [ eq_bnl ] ) below ) . figure [ pic1 ] shows a part of the the orbit @xmath54 for @xmath55 . by the result of ledrappier , this orbit is uniformly distributed in @xmath56 with respect to the measure @xmath57 . [ pic1 ]        dani and raghavan also considered orbits of frames under @xmath58 . denote @xmath59 j= (    [ cols="^,^",options="header " , ]     ) , @xmath59 and @xmath60 is a continuous function depending only on the @xmath61-components of @xmath62 . we can use proposition [ pro_assym ] with @xmath63 , @xmath64 , and @xmath65 . since @xmath34 is dense in @xmath66 , @xmath67 is dense in @xmath10 . by ( [ eq_last ] ) , the condition ( [ eq_h1 ] ) holds for @xmath63 . since @xmath68 is unipotent , the condition ( [ eq_h2 ] ) for @xmath63 holds too @xcite .    applying proposition [ pro_assym ] , we get @xmath69 as @xmath31 , where @xmath70 is defined in ( [ eq_dh ] ) . thus , by ( [ eq_last ] ) , @xmath71 where @xmath72 to find the constant @xmath73 , we calculate measures of the set @xmath74 denote by @xmath75 the lebesgue measure of a @xmath76-dimensional unit ball . recall that @xmath77 clearly , @xmath78 for @xmath79 , @xmath80 , and @xmath81 , @xmath82 iff @xmath83 for @xmath84 . we have @xmath85 let as introduce new coordinates on @xmath86 : @xmath87 , @xmath88 . the haar measure on @xmath86 ( [ eq_da ] ) is given by @xmath89 . by ( [ eq_nnn ] ) , the set of @xmath90 such that @xmath91 is described by conditions : @xmath92 thus , @xmath93 in the last step , we have used ( [ eq_vball ] ) and the well - known identity for @xmath5-function and @xmath61-function . finally , by ( [ eq_vol1 ] ) and ( [ eq_vol2 ] ) , @xmath94    let @xmath95 for @xmath96 , and @xmath97 for @xmath98 , define @xmath99 note that @xmath100 . thus , it is enough to compute asymptotics of the function @xmath101 as @xmath102 . by tauberian theorem ( v , theorem  4.3 ) , it can be deduced from asymptotics of the function @xmath103 as @xmath104 . it is more convenient to work with the function @xmath105 let @xmath106 . one can check that @xmath107 for @xmath108 . ( in fact , each of the integral defines a right haar measure on @xmath109 . ) consider mellin transform of the function @xmath110 : @xmath111 using that @xmath112 , we get @xmath113 making substitution @xmath114 , we get @xmath115 by mellin inversion formula , for sufficiently large @xmath116 , @xmath117 since @xmath5-function decays fast on vertical strips , we can shift the line of integration to the left . by ( [ eq_fz ] ) , the first pole of @xmath118 occurs at @xmath119 . therefore , it follows from ( [ eq_fff ] ) that @xmath120 by ( [ eq_psi2 ] ) , @xmath121 finally , the asymptotic estimate for @xmath122 as @xmath102 follows from tauberian theorem ( * ? ? ? * ch . v , theorem  4.3 ) . we have @xmath123 this proves the lemma . note that @xmath127 we use the formula ( [ eq_rho_l ] ) for @xmath128 and make the change of variables @xmath129 for @xmath130 . the formula ( [ eq_rbtc ] ) follows from the fact that the volume of a unit ball in @xmath131 is @xmath132 .    for @xmath133 , put @xmath134 we claim that @xmath135 as @xmath31 . if @xmath136 , then @xmath137 for every @xmath138 . then as in lemma [ lem_btc ] , @xmath139 now the claim follows from ( [ eq_btasy ] ) . since @xmath140 we have @xmath141 therefore , @xmath142 as @xmath31 .    by theorem [ th_dr ] , @xmath143 is dense in @xmath1 . by theorem [ th_frames00 ] , ( [ eq_ntasy ] ) holds . the volume of @xmath13 was computed by minkowski . for the measure @xmath144 , we have @xmath145 ( see ( * ? ? ? * theorem 5.6 ) ) . 30 s.  g.  dani , g.  a.  margulis , _ limit distributions of orbits of unipotent flows and values of quadratic forms_. i. m. gelfand seminar , 91137 , adv . soviet math . , 16 , part 1 , ams , providence , ri , 1993 .
we study distribution of orbits of a lattice @xmath0 in the the space @xmath1 of @xmath2-frames in @xmath3 ( @xmath4 ) . 
 examples of dense @xmath5-orbits are known from the work of dani , raghavan , and veech . 
 we show that dense orbits of @xmath5 are uniformly distributed in @xmath1 with respect to an explicitly described measure . 
 we also establish analogous result for lattices in @xmath6 that act on the space of isotropic @xmath7-frames .