On the distribution of $\alpha p^{\gamma}+\beta$ modulo one

TL;DR

This paper investigates the distribution of the sequence \\alpha p^{\\gamma} + \\beta modulo one for primes p with specific properties, establishing conditions under which the sequence approximates integers closely infinitely often.

Contribution

It introduces new results on the distribution of \\alpha p^{\\gamma} + \\beta modulo one for special primes, using exponential sum estimates and sieve methods.

Findings

Proves infinitely many primes p satisfy \\|\\\alpha p^{\\gamma} + \\beta \\| < p^{-\\ heta} under certain conditions.

Extends understanding of fractional parts of polynomial sequences at primes with restricted factorization.

Connects distribution results to recent advances in exponential sum estimates and sieve techniques.

Abstract

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta}, \end{equation*} holds for infinitely many primes $p$ of a special type. In particular, we consider when this inequality holds for primes $p$ such that $p+2$ has few prime factors counted with multiplicity. This is done using an exponential sum estimate of the author and the linear sieve of Iwaniec with bilinear error term. This is related to recent work of Tolev, Todorova, Mat\"{o}maki and Cai.