Diophantine approximation with primes from short intervals

TL;DR

This paper investigates how well primes within short intervals approximate irrational numbers with bounded continued fraction terms, providing asymptotic counts under specific conditions.

Contribution

It establishes new asymptotic results on the distribution of primes in short intervals approximating certain irrational numbers.

Findings

Primes in short intervals approximate quadratic irrationals with predictable frequency.

Asymptotic formula for the count of such primes is derived.

Results depend on interval length and approximation parameter.

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 $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le \delta\le 1/2$.