$6$-th Norm of a Steinhaus Chaos

TL;DR

This paper establishes the asymptotic behavior of the sixth moment of sums of Steinhaus random variables over integers with a fixed number of prime factors, revealing a cubic relation under certain conditions.

Contribution

It provides a precise asymptotic estimate for the sixth moment of Steinhaus chaos sums over integers with a fixed number of prime factors, extending understanding of their probabilistic structure.

Findings

The sixth moment of the sum scales as the cube of the set size.

The result holds for m much less than (log log N)^{1/3}.

The expectation is asymptotically comparable to |E_{N,m}|^3.

Abstract

We prove that for the Steinhaus Random Variable $z(n)$ \[\mathbb{E}\left(\left|\sum_{n\in E_{N, m}}z(n)\right|^6\right)\asymp |E_{N, m}|^3 \text{ for } m\ll(\log\log N)^{\frac{1}{3}},\] where \[E_{N, m}:=\{1\leq n:\Omega(n)=m\}\] and $\Omega(n)$ denotes the number of prime factors of $N$.