Hilbert cubes in progression-free sets and in the set of squares II

TL;DR

This paper proves that the dimension of Hilbert cubes contained within the set of perfect squares up to N is bounded by a double logarithmic function, advancing understanding of additive structures in sparse sets.

Contribution

It establishes a new upper bound on the dimension of Hilbert cubes within the squares, showing it grows at most logarithmically double in the size of the set.

Findings

Dimension of Hilbert cubes in squares is O(log log N)

Provides bounds on additive structures in sparse sets

Advances understanding of progression-free subsets in squares

Abstract

Let $S_2$ be the set of integer squares. We show that the dimension $d$ of a Hilbert cube $a_0+\{0,a_1\}+...+ \{0, a_d\}\subset S_2$ is bounded by $d=O(\log \log N)$