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...
| Үндсэн зохиолчид: | , |
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| Формат: | Journal article |
| Хэл сонгох: | English |
| Хэвлэсэн: |
Princeton University, Department of Mathematics
2020
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| Тойм: | 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. |
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