Convolution Estimates and Number of Disjoint Partitions

Paata Ivanisvili

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. 


Keywords


Clusters; Disjoint partitions; Hamming cube

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