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The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Lorenz Frühwirth, Manuel Hauke

Abstract

Given a monotonically decreasing $ψ: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $α\in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that $\left\lvert α- \frac{p}{q}\right\rvert \leq \frac{ψ(q)}{q}$. The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any $k\geq 1$, any function $ψ: \mathbb{N} \to [0,1/2]$ (not necessarily monotonic) and almost every $α\in \mathbb{R}^k$, there exist infinitely many $q$ such that $\prod\limits_{i=1}^k \left\lvert α_i - \frac{p_i}{q}\right\rvert \leq \frac{ψ(q)}{q^k}, p_1,\ldots,p_k$ all coprime to $q$, if and only if \[\sum\limits_{q \in \mathbb{N}} ψ(q) \left(\frac{\varphi(q)}{q} \right)^k\log \left(\frac{q}{\varphi(q)ψ(q)}\right)^{k-1} = \infty.\] This settles a conjecture of Beresnevich, Haynes, and Velani.

The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Abstract

Given a monotonically decreasing , Khintchine's Theorem provides an efficient tool to decide whether, for almost every , there are infinitely many such that . The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any , any function (not necessarily monotonic) and almost every , there exist infinitely many such that all coprime to , if and only if This settles a conjecture of Beresnevich, Haynes, and Velani.
Paper Structure (13 sections, 10 theorems, 105 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 105 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. The metric theory of Diophantine approximation
  3. The approximation theory in higher dimensions
  4. Applications and related work
  5. Hausdorff dimension analogue
  6. Open questions
  7. Notation
  8. Proof of the main result
  9. The convergent case and a discussion on the correct divergence condition
  10. The divergent case
  11. Proof of Theorem \ref{['main_thm']} assuming a variance estimate
  12. The variance estimate
  13. Proof of Corollaries \ref{['littlewood_subsequence']},\ref{['Hausdorff_thm']} and \ref{['Hausdorff_cor']}

Key Result

Theorem 1

Let $\psi: \mathop{\mathrm{\mathbb{N}}}\nolimits \to [0,\infty)$ be a monotonically decreasing function. Writing we have

Figures (3)

  • Figure 1: An illustration of the set $A_4$ (the non-coprime setting). The red area corresponds to $A_{4,1,1}$. Note that for illustration purposes, the lower-left corner has the coordinate $( 1/8, 1/8 )$.
  • Figure 2: To the left, we see $A_4'$, which is the proper set in the coprime setting. To the right, we have $A_4"$, which would give the conjectured divergence criterion in Beresnevich_Haynes_Velani2013. We note that in both figures, the lower left corner has the coordinate $(1/8,1/8)$.
  • Figure 3: The picture illustrates the relation of $f_q^{(k)}(y)$, $f_{q,\varepsilon}^{(k)}(y)$ and $\frac{1}{K} \sum_{i=0}^{K-1} \mathds{1}_{[-x_{q,i}, x_{q,i}]}(y)$.

Theorems & Definitions (17)

  • Theorem : Khintchine, 1924
  • Theorem : Gallagher, 1962
  • Conjecture 1
  • Conjecture 2
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Proposition 2.1
  • ...and 7 more