Diophantine approximation with primes from short intervals
Stephan Baier, Sayantan Roy
TL;DR
This work proves a hybrid result linking Diophantine approximation with primes in short intervals: for α with bounded continued-fraction terms (e.g., quadratic irrationals) and Y in a long-interval regime, the count of primes p in (X−Y,X] with ||pα||<δ is asymptotically proportional to δY/log X, under precise relations among X,Y,δ and the Diophantine denominator q. The authors smooth the small-||x|| condition via a Gaussian-type function F, apply Vaughan’s identity to split the sum into type I/II bilinear forms, and bound the resulting exponential sums using Diophantine-approximation lemmas alongside the Guth–Maynard short-interval primes result. A detailed analysis of the type I and type II sums yields sufficient decay to establish the asymptotic, provided the continued-fraction bounds on α hold and the parameters satisfy the stated inequalities. This approach bridges classical Diophantine methods with modern short-interval prime results, extending prior work on primes near multiples of irrational numbers into a rigorous hybrid setting.
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 $α$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $α$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||pα||<δ$ is asymptotically equal to $2δY/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le δ\le 1/2$.