Diophantine approximation with primes in an arithmetic progression

Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, $\beta \in \mathbb{R}$, $N\in {\mathbb{R}}_{\ge 1}$ and $\mathrm{\Delta }\in (0,1\mathrm{/}2)$. For any real $y$, let $\mathrm{\parallel }y\mathrm{\parallel }$ 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 $\mathrm{\ell }\equiv uv$ such that $\mathrm{\parallel }\alpha \mathrm{\ell }-\beta \mathrm{\parallel }{\ll }_v{\mathrm{\ell }}^{-1\mathrm{/}4}{\log }^8\mathrm{\ell }\mathrm{.}$ 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 $f_{\mathrm{\Delta }}(\theta )$ on $\mathbb{R}$ such that $f_{\mathrm{\Delta }}(\theta )$ is $1\text{ if }\mathrm{\parallel }\theta \mathrm{\parallel }<\mathrm{\Delta }$ and $0$ otherwise. Further, suppose that $u,v\in {\mathbb{Z}}_{\ge 1}$ are coprime and $a$, $q\in \mathbb{Z}$ are coprime with $q>N^{1\mathrm{/}4}$ and $\mathrm{\mid }\alpha v-a\mathrm{/}q\mathrm{\mid }\le 1\mathrm{/}{q}^2$. Then, for every $\epsilon \in {\mathbb{R}}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \ n \equiv u \bmod v}}^N Λ(n) (f_Δ(αn - β) - 2Δ) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}Δ^{1/2} + (ΔNq)^{1/2} + N^εq Δ^{1-ε}) $\mathcal{L}$^8 \end{equation*} where $\mathcal{L}=\log (Nq\mathrm{/}\mathrm{\Delta })$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N Λ(n) (f_Δ(αn - β) - 2Δ). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S\subset \mathbb{R}\setminus \text{Q}$ such that for every $\gamma \in S$ there are infinitely many primes $\mathrm{\ell }\equiv uv$ satisfying $\mathrm{\parallel }\gamma \mathrm{\ell }\mathrm{\parallel }<{\mathrm{\ell }}^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{ö}s and Mahler from 1939.