Convolution Estimates and Number of Disjoint Partitions

  • Paata Ivanisvili
Keywords: Clusters, Disjoint partitions, Hamming cube

Abstract

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate the number from above by $|X|^{c(n)}$ where
$$
c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}.
$$
This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions. 

Published
2017-06-30
Article Number
P2.43