[Paper Review] The Duffin-Schaeffer Conjecture with extra divergence

Alan Haynes, Andrew Pollington+1 more|ArXiv.org|Nov 7, 2008

Mathematical Dynamics and Fractals5 references4 citations

TL;DR

This paper establishes a strengthened version of the Duffin-Schaeffer Conjecture by introducing an 'extra divergence' condition using a slowly decaying function $ f $, proving that the set $ W(\psi) $ of well-approximable numbers has full Lebesgue measure when $ \sum f(\psi(n)/n) \varphi(n) = \infty $. It further shows that the Hausdorff dimension of $ W(\psi) $ is 1 whenever the sum diverges with $ \epsilon = 0 $, confirming the dimension analogue of the conjecture.

ABSTRACT

Given a nonnegative function $ψ: \N o \R $, let $W(ψ)$ denote the set of real numbers $x$ such that $|nx -a| < ψ(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(ψ)$ is of full Lebesgue measure if there exists an $ε> 0 $ such that $$ extstyle \sum_{n\in\N}(\frac{ψ(n)}{n})^{1+ε}φ(n)=\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement with $ε= 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(ψ)$ is of full Hausdorff dimension if the above sum with $ε= 0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.

Motivation & Objective

• To establish a sufficient condition for the set $ W(\psi) $ to have full Lebesgue measure under a strengthened divergence condition beyond the original Duffin-Schaeffer Conjecture.
• To prove the dimension analogue of the Duffin-Schaeffer Conjecture, showing that $ \dim(W(\psi)) = 1 $ whenever the sum $ \sum \psi(n)/n \cdot \varphi(n) $ diverges.
• To investigate the Duffin-Schaeffer Conjecture through weaker, more tractable conjectures involving extra divergence, aiming to gain insight toward the full conjecture.

Proposed method

• Introduces a function $ f(x) $ that decays faster than $ x(-\log x)^{-\epsilon} $ but slower than $ x^{1+\epsilon} $, and proves that divergence of $ \sum f(\psi(n)/n) \varphi(n) $ implies $ \lambda(W(\psi)) = 1 $.
• Applies the Mass Transference Principle to extend the Lebesgue measure result to Hausdorff measure and dimension, linking measure-theoretic and dimensional properties of $ W(\psi) $.
• Uses the Borel-Cantelli Lemma and estimates on pairwise intersections $ \lambda(\mathcal{E}_m \cap \mathcal{E}_n) $ to control the probability of overlapping approximations.
• Employs the function $ f $ to interpolate between known results: it generalizes Corollary 1 (with $ \epsilon > 0 $) and approaches the critical case $ \epsilon = 0 $.
• Demonstrates that the divergence condition $ \sum \psi(n)/n \cdot \varphi(n) = \infty $ implies $ \dim(W(\psi)) = 1 $, confirming the dimension analogue of the Duffin-Schaeffer Conjecture.
• Proposes Conjecture 2 and Problem 2 as natural next steps, focusing on verifying the conjecture for $ f(r) = r(\log 1/r)^{-1} $, which lies just beyond known results.

Experimental results

Research questions

• RQ1 Does the Duffin-Schaeffer Conjecture hold under an 'extra divergence' condition that strengthens the divergence beyond the classical sum $ \sum \psi(n)/n \cdot \varphi(n) = \infty $?
• RQ2 Is the dimension analogue of the Duffin-Schaeffer Conjecture true, i.e., does $ \dim(W(\psi)) = 1 $ whenever the classical sum diverges?
• RQ3 Can the Duffin-Schaeffer Conjecture be approached via weaker conjectures involving slowly decaying functions $ f $, such as $ f(r) = r(\log 1/r)^{-1} $, which lie between known cases and the critical case?
• RQ4 Does the Mass Transference Principle allow the extension of Lebesgue measure results to Hausdorff measure results for dimension functions $ h $ that are closer to Lebesgue measure than $ r^{1-\epsilon} $?
• RQ5 Can Conjecture 1 (generalized Hausdorff measure version) be verified for the dimension function $ h(r) = r \log(1/r) $, which lies just beyond the current results?

Key findings

• The paper proves that $ \lambda(W(\psi)) = 1 $ if $ \sum f(\psi(n)/n) \varphi(n) = \infty $, where $ f(x) $ decays faster than $ x(-\log x)^{-\epsilon} $ but slower than $ x^{1+\epsilon} $ for any $ \epsilon > 0 $, thus establishing a new sufficient condition for full Lebesgue measure.
• Corollary 1 shows that $ \lambda(W(\psi)) = 1 $ whenever $ \sum (\psi(n)/n)^{1+\epsilon} \varphi(n) = \infty $ for some $ \epsilon > 0 $, which is equivalent to a known result in Harman's book.
• Theorem 2 establishes that $ \dim(W(\psi)) = 1 $ if $ \sum (\psi(n)/n)^{1-\epsilon} \varphi(n) = \infty $ for all $ \epsilon > 0 $, confirming the dimension analogue of the Duffin-Schaeffer Conjecture.
• Corollary 3 confirms that $ \dim(W(\psi)) = 1 $ whenever $ \sum \psi(n)/n \cdot \varphi(n) = \infty $, which is the dimension analogue of the original Duffin-Schaeffer Conjecture.
• The paper verifies Conjecture 1 (generalized Hausdorff measure) for the dimension function $ h(r) = r^{1 - 1/(1 + \log \log 1/r)} $, which decays slower than $ r^{1-\epsilon} $ but faster than $ r \log(1/r) $, showing progress toward the full conjecture.
• Problem 2 — verifying Conjecture 2 for $ f(r) = r(\log 1/r)^{-1} $ — is proposed as the next natural step, as it lies just beyond the current result and would imply the full dimension analogue.