Sieving with square conditions and applications to Hilbert cubes in arithmetic sets

TL;DR

This paper develops new sieve techniques to study Hilbert cubes in various sets, providing bounds on their maximal dimension and improving understanding of their structure in number theory.

Contribution

It introduces methods to adapt Gallagher's sieve for prime squares and applies these to bound the dimension of Hilbert cubes in different multiplicative sets.

Findings

Bound of O(log N) for Hilbert cubes in squareful numbers

Similar bounds for multiplicative semigroups with positive prime density

Improved maximal dimension bounds for subset sums of pure powers

Abstract

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information modulo $p$, so that we can successfully apply the sieve. 2) The applications to Hilbert cubes are of interest in their own right: We study the maximal dimension of Hilbert cubes in various multiplicatively defined sets. For the squareful numbers in $[1,N]$ we achieve an upper bound of the dimension of $d=O(\log N)$. The same upper bounds follow for multiplicative semigroups of integers defined by a positive proportion of the primes, and the set of integers representable by an irreducible positive definite binary quadratic form. Eventually, making use of the sun flower lemma we give an improvement on the maximal dimension $d$ of subset sums in the set of pure powers in $[1,N]$.