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arXiv · 2512.02174

Diophantine approximation with primes from short intervals

Stephan Baier·Sayantan Roy
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Abstract

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $\alpha$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $\alpha$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||p\alpha||<\delta$ is asymptotically equal to $2\delta Y/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le \delta\le 1/2$.

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Stephan Baier, Sayantan Roy. 2025-12-01. Diophantine approximation with primes from short intervals. https://arxiv.org/abs/2512.02174

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