Diophantine approximation with primes in an arithmetic progression

Abstract

Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \|<\Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q>N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \|<\ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{\"o}s and Mahler from 1939.