Two dimensional arithmetic progressions avoiding squares

Rainer Dietmann; Christian Elsholtz

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arXiv · 2605.11104

Two dimensional arithmetic progressions avoiding squares

Rainer Dietmann·Christian Elsholtz

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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 $O_\varepsilon(T^{20/27+\varepsilon})$ 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.

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Rainer Dietmann, Christian Elsholtz. 2026-05-11. Two dimensional arithmetic progressions avoiding squares. https://arxiv.org/abs/2605.11104

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