PreprintCC BY 4.0arXiv
Families that intersect in an arithmetic progression
Abstract
For n∈ℕ, let μn be the uniform measure on 2[n]. A family ℱ ⊆2[n] is called k-AP-intersecting if F1∩ F2 contains a nonconstant k-term arithmetic progression for every F1, F2∈ℱ. Writing mk for the supremum of μn(ℱ) over all n and all such families ℱ, we prove 2−k≤ mk≤32· 2−k (k≥3). This greatly improves a recent result of Gillott, who proved that mk ≤ k−1/3 for all sufficiently large k, and answers his question on mk's exponential decay. We also prove a similar bound for the corresponding cross-intersection problem.
Related
- Some Infinite Families of Arithmetic Symplectic Hypergeometric Groups
- Geometric families of multiple elliptic Gamma functions and arithmetic applications, III
- On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture
- Diophantine approximation with primes in an arithmetic progression
- Longest cycles intersect linearly in highly connected graphs
BibTeX
@misc{kupavskii2026families,
title = {{Families that intersect in an arithmetic progression}},
author = {Andrey Kupavskii and Fedor Noskov and Georgy Sokolov and Nikolai Terekhov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26243},
eprint = {2610.26243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Families that intersect in an arithmetic progression
AU - Andrey Kupavskii
AU - Fedor Noskov
AU - Georgy Sokolov
AU - Nikolai Terekhov
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26243
UR - https://arxiv.org/abs/2610.26243
AB - For n∈ℕ, let μ_n be the uniform measure on 2^[n]. A family ℱ ⊆2^[n] is called k-AP-intersecting if F_1∩ F_2 contains a nonconstant k-term arithmetic progression for every F_1, F_2∈ℱ. Writing m_k for the supremum of μ_n(ℱ) over all n and all such families ℱ, we prove 2^−k≤ m_k≤32· 2^−k (k≥3). This greatly improves a recent result of Gillott, who proved that m_k ≤ k^−1/3 for all sufficiently large k, and answers his question on m_k's exponential decay. We also prove a similar bound for the corresponding cross-intersection problem.
ER -