On the Duffin-Schaeffer conjecture
By Dimitris Koukoulopoulos, James Maynard. Annals of Mathematics, 2019
80 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/93a009fd98469076c0a44f00724858d10aa22f9f
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- Mass transference principle from rectangles to rectangles in Diophantine approximation (2019, methodology): “Maynard [34], a Hausdorff measure version of Duffin-Schaeffer conjecture is also valid under the help of Theorem 2.”