PreprintCC BY 4.0arXiv
Hitting Arithmetic Progressions at the Square-Root Scale
Abstract
Let f(N,k) be the minimum size of a subset of {0,…,N−1} meeting every k-term arithmetic progression of positive difference contained in that interval. Improving on a bound of Truss, we prove that f(n2,n)≥ n+n1−o(1) as n→∞. Improving on a bound of Xu, for primes p we prove that f(p2,p)≤2p−(√2−o(1))√(p/(log p)).
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BibTeX
@misc{korsky2026hitting,
title = {{Hitting Arithmetic Progressions at the Square-Root Scale}},
author = {Samuel Korsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.01780},
eprint = {2606.01780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01780},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hitting Arithmetic Progressions at the Square-Root Scale
AU - Samuel Korsky
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.01780
UR - https://arxiv.org/abs/2606.01780
AB - Let f(N,k) be the minimum size of a subset of {0,…,N−1} meeting every k-term arithmetic progression of positive difference contained in that interval. Improving on a bound of Truss, we prove that f(n^2,n)≥ n+n^1−o(1) as n→∞. Improving on a bound of Xu, for primes p we prove that f(p^2,p)≤2p−(√2−o(1))√(p/(log p)).
ER -