Journal article
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
Actions
Access Document
- Files:
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(Preview, Accepted manuscript, 592.1KB, Terms of use)
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- Publisher copy:
- 10.4007/annals.2020.192.1.5
Authors
- Publisher:
- Princeton University, Department of Mathematics
- Journal:
- Annals of Mathematics More from this journal
- Volume:
- 192
- Issue:
- 2020
- Pages:
- 251-307
- Publication date:
- 2020-07-17
- Acceptance date:
- 2020-05-01
- DOI:
- EISSN:
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1939-8980
- ISSN:
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0003-486X
- Language:
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English
- Keywords:
- Pubs id:
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1102622
- Local pid:
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pubs:1102622
- Deposit date:
-
2020-05-01
Terms of use
- Copyright holder:
- Department of Mathematics, Princeton University
- Copyright date:
- 2020
- Rights statement:
- © 2020 Department of Mathematics, Princeton University
- Notes:
- This is the accepted manuscript version of the article. The final version will be available from Princeton University Press at: https://doi.org/10.4007/annals.2020.192.1.5
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