Conjecture similar to Dirichlet's approximation theorem but with prime numbers

Question

Dirichlet's approximation theorem states that for any real numbers $\alpha,N$ with $1\leq N,$ there exist integers $p$ and $q$ such that $1\leq q\leq N$ and $\lvert q\alpha-p \rvert \leq \frac{1}{\lfloor N \rfloor + 1} < \frac{1}{N}.$

I wonder if the following is true, or if not, something closely related:

For any real numbers $\alpha,N$ with $N>0,$ there exist integers $p$ and $q$ such that $q\in \lbrace{ \text{ first } \lceil N \rceil \text{ primes } \rbrace}$ and $\lvert q\alpha-p \rvert < \frac{1}{N}.$

I have failed to find a counter-example, but I also don't have many great ideas to prove it true. In particular, I doubt we can use the pigeon-hole principle proof of Dirichlet's theorem here, but maybe I am wrong about this?

I also don't see how Minkowski's theorem or the generalisations to Dirichlet's theorem can help here.

Answer 1

"In the thirties of the last century, I. M. Vinogradov proved that the inequality $\|p\alpha\|\le p^{-1/5+\epsilon}$ has infinitely prime solutions $p$, where $\|.\|$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $\|p\alpha\|\le p^{-1/3+\epsilon}$. This exponent $1/3$ is considered the limit of the current technology."

This is taken from the abstract of Stephan Baier and Esrafil Ali Molla, Diophantine approximation with prime denominator in real quadratic function fields, https://arxiv.org/abs/2302.01717

Answer 2

I agree with Gerry Myerson. Our expectation is that $|q\alpha-p|<1/q$ for infinitely many prime numbers $q$. It follows from Linnik’s conjecture on the least prime in an arithmetic progression. However, according to a result by Harman, we don't yet know how to do better than $|q\alpha-p|<c(\log q)/(q \log \log q)$ (see https://doi.org/10.1112/plms/s3-72.2.241). Matomäki (2009) establsihed $|q\alpha-p|<q^{-1/3+\epsilon}$ unconditionally without any assumptions like the Generalized Riemann Hypothesis (GRH).