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Hilbert cubes in sets with arithmetic properties

Ernie Croot, Junzhe Mao, Chi Hoi Yip

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, Erdős-Sárközy-Stewart, Hajdu, and Sárközy, 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.

Hilbert cubes in sets with arithmetic properties

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, Erdős-Sárközy-Stewart, Hajdu, and Sárközy, 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 -th powers in an arithmetic progression for each , assuming the ABC conjecture.
Paper Structure (28 sections, 51 theorems, 181 equations)

This paper contains 28 sections, 51 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. The first framework
  3. The second framework
  4. Applications of new frameworks
  5. Applications of new frameworks to Hilbert cubes in arithmetic sets
  6. Hilbert cubes in powerful numbers
  7. Hilbert cubes in perfect powers
  8. Hilbert cubes in smooth numbers
  9. Other applications: Hilbert cubes in primes, squarefree numbers
  10. The first framework: arithmetic progressions and sumsets
  11. Proof of Theorem \ref{['main_strong']}(a)
  12. Proof of Theorem \ref{['main_strong']}(b) and (c)
  13. Hilbert cubes in powerful numbers
  14. A local version
  15. Proof of Proposition \ref{['prop:local']}
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $S\subset \mathbb{N}$ and $k\geq 2$ be a positive integer. For each $N\in \mathbb{N}$, define the following: If $a_0$ is a nonnegative integer, and $A \subseteq \mathbb{N}$ is a multiset such that $a_0+\Sigma^*(A)\subseteq S\cap [N]$, then

Theorems & Definitions (99)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 89 more