Linear equations in Piatetski-Shapiro primes
Abstract
We establish discorrelation estimates between the Piatetski-Shapiro prime set 𝒫γ := {p is prime and p = ⌊ n1/γ ⌋ for some n ∈ ℕ} and arbitrary nilsequences when γ∈ (0,1) is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in 𝒫γ to any ``finite-complexity" system of linear equations, including for the number of k-term arithmetic progressions in 𝒫γ up to a threshold N for any given k ≥ 3. Furthermore, we show that there exists an absolute constant C>0 such that if 1 − 2−Ck < γ< 1, then the Piatetski-Shapiro primes 𝒫γ contain infinitely many non-trivial k-term arithmetic progressions. This significantly improves upon the previous range of γ obtained by Li and Pan, which is of triple exponential type.
Related
- Polynomial configurations and pointwise averages along Piatetski--Shapiro primes
- Multiplicative recurrence of Piatetski--Shapiro and nil-Bohr sets
- Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
- On a variant of the Piatetski-Shapiro prime number theorem
- Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences
BibTeX
@misc{shao2026linear,
title = {{Linear equations in Piatetski-Shapiro primes}},
author = {Xuancheng Shao and Yu-Chen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.18676},
eprint = {2605.18676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.18676},
}
RIS
TY - UNPB
TI - Linear equations in Piatetski-Shapiro primes
AU - Xuancheng Shao
AU - Yu-Chen Sun
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2605.18676
UR - https://arxiv.org/abs/2605.18676
AB - We establish discorrelation estimates between the Piatetski-Shapiro prime set 𝒫_γ := {p is prime and p = ⌊ n^1/γ ⌋ for some n ∈ ℕ} and arbitrary nilsequences when γ∈ (0,1) is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in 𝒫_γ to any ``finite-complexity" system of linear equations, including for the number of k-term arithmetic progressions in 𝒫_γ up to a threshold N for any given k ≥ 3. Furthermore, we show that there exists an absolute constant C>0 such that if 1 − 2^−Ck < γ< 1, then the Piatetski-Shapiro primes 𝒫_γ contain infinitely many non-trivial k-term arithmetic progressions. This significantly improves upon the previous range of γ obtained by Li and Pan, which is of triple exponential type.
ER -