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.
Highlights
Let {0, 1}m be the Hamming cube of dimension m 1
For each fixed n exponent pn is the best possible in the sense that it cannot be replaced by any larger number
The corollary extends a recent result of Kane–Tao [1], corresponding to the case n = 3 where
Summary
Let {0, 1}m be the Hamming cube of dimension m 1. Take a finite number of functions f1, . Fn : {0, 1}m → R, and define the convolution at the corner 1m as f1 ∗ f2 ∗ . For each fixed n exponent pn is the best possible in the sense that it cannot be replaced by any larger number. As an immediate application we obtain the following corollary (see Section 2.3 below). N j=1 where denotes the disjoint union, and |X| denotes cardinality of the set. The corollary extends a recent result of Kane–Tao [1], corresponding to the case n = 3 where
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