Random unconditional convergence of Rademacher chaos in $L_\infty$ and sharp estimates for discrepancy of weighted graphs and hypergraphs

TL;DR

This paper proves that Rademacher systems and chaos are randomly unconditionally convergent in $L_$, enabling sharp estimates for discrepancy in weighted graphs and hypergraphs, extending classical unweighted results.

Contribution

It establishes the random unconditional convergence of Rademacher chaos in $L_$ and derives sharp discrepancy estimates for weighted graphs and hypergraphs, extending classical theorems.

Findings

Rademacher chaos has random unconditional convergence in $L_$.

Sharp two-sided discrepancy estimates for weighted graphs and hypergraphs.

Extension of Erd"os-Spencer theorem to weighted cases.

Abstract

We prove that both multiple Rademacher system and Rademacher chaos possess the property of random unconditional convergence in the space $L_\infty$. This fact combined with some intimate connections between $L_\infty$-norms of linear combinations of elements of these systems and some special norms of matrices of their coefficients allows us to establish sharp two-sided estimates for the discrepancy of edge-weighted graphs and hypergraphs. Some of these results extend the classical theorem proved by Erd\"os and Spencer for the unweighted case.