Convolution estimates and number of disjoint partitions (Q2363105)

Summary: 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.