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

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.