Two dimensional arithmetic progressions avoiding squares

Abstract

We show that any proper symmetric two dimensional arithmetic progression contained in the interval $[-T,T]$ which avoids non-zero perfect squares has at most $O_{\epsilon }(T^{20/27+\epsilon })$ elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime.

10 pages, to appear in Proc. Amer. Math. Soc