A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes

Mengdi Wang

A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes

Number Theory 2024-10-15 v4 Combinatorics

Authors: Mengdi Wang

Abstract

Let $k\geq1$ be a fixed integer, and $\mathcal P_N$ be the set of primes no more than $N$ . We prove that if a set $\mathcal A\subset\mathcal P_N$ contains no patterns $p_1,p_1+(p_2-1)^k$ , where $p_1,p_2$ are prime numbers, then $\frac{|\mathcal A|}{|\mathcal P_N|}\ll(\log\log N)^{-\frac{1}{4k^3+23k^2}}.$

Keywords

prime number theory prime numbers arithmetic progressions

Cite

@article{arxiv.2102.11441,
  title  = {A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes},
  author = {Mengdi Wang},
  journal= {arXiv preprint arXiv:2102.11441},
  year   = {2024}
}

Comments

the final version, incorporating the referee's numerous helpful comments and corrections

A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes

Abstract

Keywords

Cite

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