Prime multiples with near-optimal Diophantine approximation

Establish that for every irrational real number \(\alpha\) and every \(\epsilon>0\), there exist infinitely many primes \(\ell\) such that \(\|\alpha\ell\|<\ell^{-1+\epsilon}\).

Background

The paper studies how well irrational multiples of primes can approximate integers, including when the primes are restricted to a fixed arithmetic progression. It proves an unconditional result with exponent −1/4-1/4, up to logarithmic factors, for primes ℓ≡u(modv)\ell\equiv u\pmod v, and it constructs irrational numbers for which the stronger exponent −1-1 occurs infinitely often along suitable primes.

The stated conjecture asks for the exponent −1+ϵ-1+\epsilon for every irrational α\alpha, which is arbitrarily close to the natural reciprocal-scale threshold. The paper does not resolve this conjecture; its introductory results instead document the best known general bounds and establish a weaker unconditional estimate.

References

It is conjectured that for every \epsilon \in \mathbb{R}_{>0}, there are infinitely many primes \ell such that

| \alpha \ell | < \ell{-1 +\epsilon}.

— Diophantine approximation with primes in an arithmetic progression  (2609.10010 - Mazumder et al., 9 Sep 2026) in Abstract; reiterated in Section 1, Introduction and statements of the results