I am interested in a sharp bound on the largest possible size $e_3({\boldsymbol{Z}_n})$ of a subset $S \in \boldsymbol{Z}_n$ such that for any three distinct elements $a, b, c \in S$ we have $a+b \not= 2c$.

The best upper and lower bounds I could find through usual means such as Mathscinet were by [T. Sanders, *Ann. of Math.* **174** (2011) 619–636][1] and [M. Elkin, *Proc. SODA* (2010) 886–905][2] respectively, where a non-modulo version is considered, i.e., they gave bounds on the largest possible size $e_3({N})$ of a subset $S \in N =\{1,2,\dots,n\}$ such that for any three distinct integers $a, b, c \in S$ it holds that $a+b \not= 2c$.

Any $S$ containing no $3$-term APs in the modulo $n$ sense is automatically a no-$3$-term-AP subset in the sense of the non-modulo $n$ version, so we have $e_3({\boldsymbol{Z}_n}) \leq e_3({N})$. The converse may not be true, but apparently we have $e_3({N}) \leq e_3({\boldsymbol{Z}_{2n}})$. So asymptotically speaking, $e_3({\boldsymbol{Z}_n})$, which I want to know, behaves pretty much the same way as $e_3({N})$. But to my layman eye, the actual values of these two may have an ever-so-slight difference.

My question is, how large can this "same difference" be? Is there serious research somewhere exactly on this?

I'm sorry if I'm missing something quite obvious; the inequalities $e_3({\boldsymbol{Z}_n}) \leq e_3({N}) \leq e_3({\boldsymbol{Z}_{2n}})$ look too elementary.

  [1]: http://annals.math.princeton.edu/2011/174-1/p20
  [2]: http://dl.acm.org/citation.cfm?id=1873673