On the Duffin-Schaeffer conjecture (1907.04593v3)

Abstract: Let $\psi:\mathbb{N}\to\mathbb{R}{\ge0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le \psi(q)/q$. If $\sum{q=1}^\infty \psi(q)\phi(q)/q=\infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha - a/q|\le \psi(q)/q$, giving a refinement of Khinchin's Theorem.

Insights on the Duffin-Schaeffer Conjecture

The manuscript by Dimitris Koukoulopoulos and James Maynard presents a significant achievement in number theory by resolving the Duffin-Schaeffer conjecture, a longstanding problem in the area of metric Diophantine approximation. This conjecture, originating from the work of Duffin and Schaeffer in 1941, aimed to identify conditions under which a set of real numbers can be approximated infinitely often by reduced rational fractions within given bounds, as defined by a function ψ : N → R >0.

Historically, Duffin and Schaeffer conjectured that should the series $\sum \frac{\varphi(q) \psi(q)}{q}$ diverge, then for almost every real number α, there are infinitely many reduced fractions a/q satisfying $|\alpha - a/q| < \psi(q)/q$. This conjecture mirrors Khinchin's theorem in nature, where the divergence of a similar series determines the Lebesgue measure of numbers infinitely well-approximated by rational fractions.

The main result of this paper, referred to as Theorem 1, confirms the Duffin-Schaeffer conjecture entirely under the condition that the series diverges. The authors accomplish this by intricate advancements on "GCD graphs" techniques, which innovatively manage dependencies on such graphs' structures—an essential aspect of handling problems containing significant entropy and interaction.

Methodological Insights

The approach taken by Koukoulopoulos and Maynard involves a sophisticated technique that constructs an intricate understanding of how vertices in these GCD graphs connect through edges that respect the number-theoretic condition in hand. They leverage compression arguments—conceptually similar in nature to density increment strategies by Roth—aimed at incrementally improving the structural understanding of common divisors in these graphs.

A pivotal component in their proof resides in managing specific weights on vertices using Euler's totient function, $\varphi(q)/q$. This carefully balances certain divergences involving small prime factors, preserving the efficacy of the approximating series within the investigated domain. Moreover, the authors demonstrated that without incorporating these particular weights, the strategy would not suffice—underscored by invalidations such as the counter-examples revealed had they used purely count-based measures.

Implications and Future Directions

This resolution starkly impacts the broader field of Diophantine approximation, substantially enhancing our understanding of how random algebraic structures intersect dense number sets. The corollary implications specify, with rigorous conditions, the instances of full and null Lebesgue measures in rational approximation scenarios, complementing the classical results described through Khinchin's theorem.

Furthermore, establishing links between density properties of number sets and their measure-based characteristics offers new avenues for extending these methodologies to other unsolved conjectures or related problems where independence of elements might initially seem elusive.

Conclusion

Koukoulopoulos and Maynard's final resolution of the Duffin-Schaeffer conjecture marks a pivotal advancement in the context of rational approximation theory. Their novel use of GCD graphs offers a paradigm shift in thinking about approximative sets under divergence conditions. Mathematically, their findings furnish a complete narrative for approximation paradigms connected to rational fractions and usher in further exploration into related domains of number theory and beyond. Their approach, while technical and intricately layered, remains a testament to inventive methodologies capable of tackling some of the nuanced problems within pure mathematics.