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Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
Abstract
In this paper, we prove that for every integer k≥2 and every c>1 sufficiently close to $1$, there is κ>0 such that every sufficiently large subset of {1,…,N} of density at least (loglog N)−κ contains x, x+⌊ nc⌋, x+⌊ nc⌋2, …, x+⌊ nc⌋k. We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.
Related
- Polynomial configurations and pointwise averages along Piatetski--Shapiro primes
- Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences
- Pointwise convergence of bilinear ergodic averages for commuting transformations
- Pointwise Convergence of Random Ergodic Averages at the L1 Endpoint
- Pointwise convergence of noncommutative ergodic averages along the primes
BibTeX
@misc{mousavi2026polynomial,
title = {{Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences}},
author = {Hamed Mousavi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01847},
eprint = {2609.01847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01847},
}
RIS
TY - UNPB
TI - Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
AU - Hamed Mousavi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01847
UR - https://arxiv.org/abs/2609.01847
AB - In this paper, we prove that for every integer k≥2 and every c>1 sufficiently close to $1$, there is κ>0 such that every sufficiently large subset of {1,…,N} of density at least (loglog N)^−κ contains x, x+⌊ n^c⌋, x+⌊ n^c⌋^2, …, x+⌊ n^c⌋^k. We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.
ER -