Hilbert cubes in arithmetic sets

TL;DR

This paper establishes upper bounds on the size of Hilbert cubes within various arithmetic sets like squares and powerful numbers, advancing understanding of their combinatorial structure.

Contribution

It provides new upper bounds on the maximal dimension of Hilbert cubes in sets of arithmetic interest, such as squares and powerful numbers.

Findings

Upper bounds on Hilbert cube dimensions in squares

Upper bounds in powerful numbers

Results applicable to pure powers

Abstract

We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.