The exponent of harmonic LCM avoidance

Abstract

Fix $k\ge 3$, 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 $f_k(N)=(\log N{)}^{{\gamma }_k+o(1)}$ for a well-defined exponent ${\gamma }_k\in (0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for ${\gamma }_k$ 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 \[ γ_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives $0.438899\dots <{\gamma }_3\le 0.889881\dots$.

9 pages