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Erdős's integer dilation approximation problem and GCD graphs

Dimitris Koukoulopoulos, Youness Lamzouri, Jared Duker Lichtman

Abstract

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{α\in\mathcal{A}\cap[1,x]}\frac{1}α>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(α, β)\in \mathcal{A}^2$ such that $α\neq β$ and $|nα-β| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erdős from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.

Erdős's integer dilation approximation problem and GCD graphs

Abstract

Let be a countable set such that . We prove that, for every , there exist infinitely many pairs such that and for some positive integer . This resolves a problem of Erdős from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.

Paper Structure

This paper contains 24 sections, 34 theorems, 253 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Ideas from the proof
  4. Strategy and key ingredients of the proof of Theorem \ref{['thm:solutions']}
  5. Initial maneuvers
  6. The second moment method
  7. The problem with the sets $\mathcal{M}_\alpha$
  8. Correlation estimates
  9. Constructing the set $\mathcal{A}'$
  10. The last key result towards Theorem \ref{['thm:solutions']}
  11. The overlap estimate for rational ratios
  12. Auxiliary results
  13. Proof of Lemma \ref{['lem:overlap estimate for rational ratios']}
  14. Deduction of Lemmas \ref{['lem:overlap estimate for ratios with large R']} and \ref{['lem:overlap estimate for rational ratios of typical rationals']} from Lemma \ref{['lem:overlap estimate for rational ratios']}
  15. A refinement of Behrend's theorem
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{A}\subset\mathbb{R}_{>0}$ be a discrete set such that Then, for every $\varepsilon>0$, there exists a pair $(\alpha, \beta)\in \mathcal{A}^2$ such that $\alpha\neq \beta$ and $|n\alpha-\beta| <\varepsilon$ for some positive integer $n$.

Figures (1)

  • Figure 1: Edges in a structured GCD graph.

Theorems & Definitions (91)

  • Theorem 1
  • Remark
  • Definition 1.1: primitive set
  • Remark
  • Lemma 2.1
  • Definition 2.2: Negatively correlated sets
  • Definition 2.3: Height of a real number
  • Definition 2.4: Bracket of two real numbers
  • Lemma 2.5
  • proof
  • ...and 81 more