A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal cardinality. 


There is a regularly updated website on nonaveraging sets records at <http://www.math.uni.wroc.pl/~jwr/non-ave/DATABASE.TXT>. Most of the research done on upper bounds for nonaveraging subsets of $\lbrace 1,2, \ldots ,n \rbrace$ (by Roth, Bourgain, Gowers, Tao, Green and others) involves randomness one way or another (be it in the form of Fourier analysis, extremal graph theory or ergodic theory) , and Behrend's lower bound is nonconstructive.


Despite the randomness, the optimal nonaveraging sets display some structure :


When $n=20$, there are two optimal solutions, which are $B \cup (B+5) \cup \lbrace 18 \rbrace$ and $B' \cup (B'+5) \cup \lbrace 3 \rbrace$, where $B=\lbrace 1,2,9,15 \rbrace$ and $B'=\lbrace 1,7,14,15 \rbrace$. When $n=30$, there is a unique optimal solution, $B \cup (B+19)$, where $B=\lbrace 1,3,4,8,9,11 \rbrace$. Looking at larger examples from the abovementioned website, the decomposition "two copies+error term" seems to persist, which inspires me the following list of (increasingly strong) conjectures :


\*\* Conjecture 1. \*\* There is a function $E(n)$ tending to $+\infty$ when $n$ tends to infinity, such that for any optimal nonaveraging subset $X$ of $\lbrace 1,2, \ldots ,n \rbrace$ we can write $X=B \cup (B+t) \cup C$ for some nonzero integer $t$ and some $B,C \subseteq \lbrace 1,2, \ldots ,n \rbrace$ and $|B| \geq E(n)$.


\*\* Conjecture 2. \*\* There is a function $\varepsilon(n)$ tending to $0$ when $n$ tends to infinity, such that for any optimal nonaveraging subset $X$ of $\lbrace 1,2, \ldots ,n \rbrace$ we can write $X=B \cup (B+t) \cup C$ for some nonzero integer $t$ and some $B,C \subseteq \lbrace 1,2, \ldots ,n \rbrace$ with $|C| \leq n\varepsilon(n)$.


\*\* Conjecture 3. \*\* There is a function $\varepsilon(n)$ tending to $0$ when $n$
tends to infinity, such that for any optimal nonaveraging subset $X$ of $\lbrace 1,2, \ldots ,n \rbrace$ we can write 
$X=B \cup (B+t) \cup C$ for some nonzero integer $t$ and some $B,C \subseteq \lbrace 1,2, \ldots ,n \rbrace$ with $|C| \leq |X|\varepsilon(n)$.


Note that the two copies $B$ and $B+t$ are necessarily disjoint since
$X$ is nonaveraging. Also, for each conjecture we have a weaker variant
where "any optimal $X$" is replaced with "at least one optimal $X$".


Conjectures 2 and 3 may be out of reach but conjecture 1 seems really easier because containing no two copies of a set of size at least $k$ is a much stronger requirement than being nonavering, so that the corresponding optimal sets should be much smaller. Can anyone supply a proof? 



Structure of nonaveraging sets of integers
As to Conjecture 3, I strongly doubt it is true. If it were true, one could have decomposed any optimal progression-free subset of $[1,n]$ as $B\cup(B+t)\cup C$, where $|C|$ is small as compared to $|B|$. Now, $B$ is a progression-free subset of $[1,n-t]$ with the property that $t\notin B-B$ and $2t\notin\pm(B+B-2\ast B)$, with $2\ast B:=\{2b\colon b\in B\}$. My feeling is that these requirements are quite restrictive, forcing $|B|$ to be much smaller than (roughly) one half of the size of an optimal progression-free subset of $[1,n]$.


