On the Duffin-Schaeffer conjecture
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 answe...
| Main Authors: | , |
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| פורמט: | Journal article |
| שפה: | English |
| יצא לאור: |
Princeton University, Department of Mathematics
2020
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| _version_ | 1826274212364419072 |
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| author | Koukoulopoulos, D Maynard, J |
| author_facet | Koukoulopoulos, D Maynard, J |
| author_sort | Koukoulopoulos, D |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T22:40:01Z |
| format | Journal article |
| id | oxford-uuid:5b3c5a12-c93b-4cdb-955b-63be2c9fd70f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:40:01Z |
| publishDate | 2020 |
| publisher | Princeton University, Department of Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:5b3c5a12-c93b-4cdb-955b-63be2c9fd70f2022-03-26T17:20:46ZOn the Duffin-Schaeffer conjectureJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5b3c5a12-c93b-4cdb-955b-63be2c9fd70fEnglishSymplectic ElementsPrinceton University, Department of Mathematics2020Koukoulopoulos, DMaynard, JLet 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. |
| spellingShingle | Koukoulopoulos, D Maynard, J On the Duffin-Schaeffer conjecture |
| title | On the Duffin-Schaeffer conjecture |
| title_full | On the Duffin-Schaeffer conjecture |
| title_fullStr | On the Duffin-Schaeffer conjecture |
| title_full_unstemmed | On the Duffin-Schaeffer conjecture |
| title_short | On the Duffin-Schaeffer conjecture |
| title_sort | on the duffin schaeffer conjecture |
| work_keys_str_mv | AT koukoulopoulosd ontheduffinschaefferconjecture AT maynardj ontheduffinschaefferconjecture |