Metric Diophantine approximation with primes on irrational lines

Abstract

Let c>1 be irrational. We prove that, for almost every x>0 and every ε>0, there are infinitely many triples (p,r,s), with p and r prime and s an integer, for which 0<px−r<p^−1/3+ε, 0<pcx−s<p^−1/3+ε. This improves the exponent 1/5 in the homogeneous-line case of a theorem of Baier and Ghosh. The arithmetic input is the quantitative prime-counting lower bound in Matomäki's proof concerning primes in shrinking neighborhoods of an irrational rotation. We obtain a local first-moment lower bound by fixing the prime numerator and applying the prime number theorem to the admissible prime denominators. Two Selberg sieve squares, together with explicit trigonometric interval approximations, give a pointwise upper bound whose error tends to zero in L¹ after normalization. A truncation and density-point argument then yields the almost-everywhere conclusion.

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BibTeX RIS

BibTeX

@misc{jiang2026metric,
  title = {{Metric Diophantine approximation with primes on irrational lines}},
  author = {Zirui Jiang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36380},
  eprint = {2610.36380},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36380},
}

RIS

TY  - UNPB
TI  - Metric Diophantine approximation with primes on irrational lines
AU  - Zirui Jiang
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36380
UR  - https://arxiv.org/abs/2610.36380
AB  - Let c>1 be irrational. We prove that, for almost every x>0 and every ε>0, there are infinitely many triples (p,r,s), with p and r prime and s an integer, for which 0<px−r<p^−1/3+ε, 0<pcx−s<p^−1/3+ε. This improves the exponent 1/5 in the homogeneous-line case of a theorem of Baier and Ghosh. The arithmetic input is the quantitative prime-counting lower bound in Matomäki's proof concerning primes in shrinking neighborhoods of an irrational rotation. We obtain a local first-moment lower bound by fixing the prime numerator and applying the prime number theorem to the admissible prime denominators. Two Selberg sieve squares, together with explicit trigonometric interval approximations, give a pointwise upper bound whose error tends to zero in L^1 after normalization. A truncation and density-point argument then yields the almost-everywhere conclusion.
ER  -

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