Note on the sumset of squares

TL;DR

The paper addresses lower bounding the size of the sumset $Q+Q$ when $Q$ is a finite subset of the squares and proves $|Q+Q| \ge C'|Q| (\log|Q|)^{1/3+o(1)}$; the approach combines Chang's version of Freiman's theorem with Bombieri–Zannier bounds on squares in arithmetic progressions to relate $|Q+Q|/|Q|$ to $|Q|$ via a progression cover. It situates the result in the context of Ruzsa's conjecture and related conditional bounds, and clarifies dependencies with Mei-Chu Chang's prior work. The contribution provides a self-contained outline of the argument and highlights how additive combinatorics tools apply to the set of squares, illustrating connections to incidence bounds and potential extensions. This advances understanding of sumset growth for structured sets and suggests avenues for conditional improvements and generalizations, with implications for related conjectures in additive number theory.

Abstract

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $. This result essentially is proved -- with the same tools -- by Mei-Chu Chang. See in J. Funct. Anal. 207 (2004), no 2, 444-460. So the author will withdraw this ArXiv file.

Theorems & Definitions (1)

• proof : Proof of lemma \ref{['2.2']}