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On the Duffin-Schaeffer conjecture

Abstract:
Let ψ : N → R>0 be an arbitrary function from the positive integers to the nonnegative reals. Consider the set A of real numbers α for which there are infinitely many reduced fractions a/q such that |α − a/q| 6 ψ(q)/q. If P∞ q=1 ψ(q)ϕ(q)/q = ∞, we show that 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 |α − a/q| 6 ψ(q)/q, giving a refinement of Khinchin’s Theorem.
Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.4007/annals.2020.192.1.5

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Sub department:
Mathematical Institute
Oxford college:
Magdalen College
Role:
Author
ORCID:
0000-0001-5782-7082
Publisher:
Princeton University, Department of Mathematics Publisher's website
Journal:
Annals of Mathematics Journal website
Volume:
192
Issue:
2020
Pages:
251-307
Publication date:
2020-07-17
Acceptance date:
2020-05-01
DOI:
EISSN:
1939-8980
ISSN:
0003-486X
Language:
English
Keywords:
Pubs id:
1102622
Local pid:
pubs:1102622
Deposit date:
2020-05-01

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