Hilbert cubes in sets with arithmetic properties

TL;DR

This paper develops new methods to estimate the size of Hilbert cubes within various arithmetic sets, improving existing bounds and providing conditional results based on the ABC conjecture.

Contribution

It introduces general frameworks for Hilbert cube estimation and sharpens bounds in several classical arithmetic sets, also offering conditional bounds assuming the ABC conjecture.

Findings

Sharp bounds for Hilbert cubes in perfect powers and primes

Conditional upper bounds on k-th powers in arithmetic progressions

Improved estimates for Hilbert cubes in various number sets

Abstract

In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets, including perfect powers, powerful numbers, primes, smooth numbers, and squarefree numbers. Along the way, we substantially sharpen several earlier results of Dietmann-Elshotlz, Erd\H{o}s-S\'ark\"ozy-Stewart, Hajdu, and S\'ark\"ozy, and we obtain bounds that are sharp up to the implied constant in several cases. Additionally, we prove conditional results of independent interest, including an almost sharp uniform upper bound on the number of $k$-th powers in an arithmetic progression for each $k\geq 4$, assuming the ABC conjecture.