PreprintarXiv licencearXiv
The exponent of harmonic LCM avoidance
Abstract
Fix k≥ 3, and let fk(N) be the largest harmonic sum of a subset of [N] containing no k distinct integers with a common pairwise least common multiple. We prove that fk(N)=(log N)γk+o(1) for a well-defined exponent γk∈(0,1]. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for γk in terms of weighted sunflower-free families. We then eliminate the continuous weight: if Mk(n,r) is the largest size of an r-uniform k-cosunflower-free family on [n], then γk=supn≥1, 1≤ r≤ nr/enMk(n,r)1/r. This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives 0.438899…<γ3≤0.889881….
Related
- Return-Growth Sequences: Euler Enumeration and Pattern Avoidance
- Pattern avoidance in alternating sign rectangles I: Extended avoidance
- Pattern avoidance classes of parking permutations with fixed lucky sets
- Staircase diagrams, pattern avoidance, and smooth permutations
- Cone Avoidance and the No-Least-Join Theorem
BibTeX
@misc{luo2026exponent,
title = {{The exponent of harmonic LCM avoidance}},
author = {Yanping Luo and Ruiyi Yang and Keheng Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07268},
eprint = {2609.07268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07268},
}
RIS
TY - UNPB
TI - The exponent of harmonic LCM avoidance
AU - Yanping Luo
AU - Ruiyi Yang
AU - Keheng Zhu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07268
UR - https://arxiv.org/abs/2609.07268
AB - Fix k≥ 3, and let f_k(N) be the largest harmonic sum of a subset of [N] containing no k distinct integers with a common pairwise least common multiple. We prove that f_k(N)=(log N)^γ_k+o(1) for a well-defined exponent γ_k∈(0,1]. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for γ_k in terms of weighted sunflower-free families. We then eliminate the continuous weight: if M_k(n,r) is the largest size of an r-uniform k-cosunflower-free family on [n], then γ_k=sup_n≥1, 1≤ r≤ nr/enM_k(n,r)^1/r. This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives 0.438899…<γ_3≤0.889881….
ER -