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Further bounds in the polynomial Szemerédi theorem
Abstract
We show that there exists c > 0 such that any subset of {1,…,N} having size ≫ N / exp( (logloglog N)c ) contains a nontrivial pattern of the form x,x+y,x+2y,x+y3. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.
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BibTeX
@misc{bychawski2026further,
title = {{Further bounds in the polynomial Szemerédi theorem}},
author = {Bartłomiej Bychawski and Bartosz Głowacki and Ivan Spyrydonov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12032},
eprint = {2610.12032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12032},
}
RIS
TY - UNPB
TI - Further bounds in the polynomial Szemerédi theorem
AU - Bartłomiej Bychawski
AU - Bartosz Głowacki
AU - Ivan Spyrydonov
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12032
UR - https://arxiv.org/abs/2610.12032
AB - We show that there exists c > 0 such that any subset of {1,…,N} having size ≫ N / exp( (logloglog N)^c ) contains a nontrivial pattern of the form x,x+y,x+2y,x+y^3. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.
ER -