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Diophantine approximation with smooth numbers

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Abstract

Let \(\theta \) be an irrational number and \(\varphi \) a real number. Let \(C > 2\) and \(\varepsilon > 0\). There are infinitely many positive integers n free of prime factors \(> (\log n )^C\), such that \(\Vert \theta n + \varphi \Vert < n^{-\left( \frac{1}{3} - \frac{2}{3C}\right) + \varepsilon }\). Here \(\Vert y\Vert \) is the distance from y to \(\mathbb Z\).

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Roger Baker

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  1. Roger Baker
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Correspondence to Roger Baker.

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In memory of Richard Askey

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Baker, R. Diophantine approximation with smooth numbers. Ramanujan J 61, 49–54 (2023). https://doi.org/10.1007/s11139-020-00361-z

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  • DOI: https://doi.org/10.1007/s11139-020-00361-z

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