Diophantine approximation with primes in an arithmetic progression
Abstract
Let α∈ ℝ ∖ ℚ, β∈ R, N ∈ ℝ≥ 1 and $ Δ\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 ≥ 1, there are infinitely many primes ℓ ≡ u bmod v such that $ \|α\ell - β\| \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 fΔ(θ) on R such that fΔ(θ) is $ 1 \text{ if } \| θ\| < Δ$ and $ 0 $ otherwise. Further, suppose that u, v ∈ ℤ≥ 1 are coprime and a, q ∈ Z are coprime with q> N1/4 and |αv −a/q| ≤ 1/q2. Then, for every ε∈ ℝ>0 we have ∑substackn=1 n ≡ u bmod vN Λ(n) (fΔ(αn − β) − 2Δ) ≪v (Nq−1/2 + N3/4 + N5/6Δ1/2 + (ΔNq)1/2 + Nεq Δ1−ε) ℒ8 where ℒ=log (Nq/Δ). This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum ∑substackn=1N Λ(n) (fΔ(αn − β) − 2Δ). In the second part of this paper, we explicitly construct an uncountable set S ⊂ R∖ Q such that for every γ∈ S there are infinitely many primes ℓ ≡ u bmod v satisfying | γℓ | < ℓ−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.
Related
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BibTeX
@misc{mazumder2026diophantine,
title = {{Diophantine approximation with primes in an arithmetic progression}},
author = {D. Mazumder and J. Sivaraman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10010},
eprint = {2609.10010},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10010},
}
RIS
TY - UNPB
TI - Diophantine approximation with primes in an arithmetic progression
AU - D. Mazumder
AU - J. Sivaraman
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10010
UR - https://arxiv.org/abs/2609.10010
AB - Let α∈ ℝ ∖ ℚ, β∈ R, N ∈ ℝ_≥ 1 and $ Δ\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 ≥ 1, there are infinitely many primes ℓ ≡ u bmod v such that $ \|α\ell - β\| \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_Δ(θ) on R such that f_Δ(θ) is $ 1 \text{ if } \| θ\| < Δ$ and $ 0 $ otherwise. Further, suppose that u, v ∈ ℤ_≥ 1 are coprime and a, q ∈ Z are coprime with q> N^1/4 and |αv −a/q| ≤ 1/q^2. Then, for every ε∈ ℝ_>0 we have ∑_substackn=1 n ≡ u bmod v^N Λ(n) (f_Δ(αn − β) − 2Δ) ≪_v (Nq^−1/2 + N^3/4 + N^5/6Δ^1/2 + (ΔNq)^1/2 + N^εq Δ^1−ε) ℒ^8 where ℒ=log (Nq/Δ). This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum ∑_substackn=1^N Λ(n) (f_Δ(αn − β) − 2Δ). In the second part of this paper, we explicitly construct an uncountable set S ⊂ R∖ Q such that for every γ∈ S there are infinitely many primes ℓ ≡ u bmod v satisfying | γℓ | < ℓ^−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.
ER -