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Random unconditional convergence of Rademacher chaos in L_∞ and sharp estimates for discrepancy of weighted graphs and hypergraphs · arXivDesk

2412.20107Dec 28, 202430 pages

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

Sergey V. Astashkin, Konstantin V. Lykov

Probability

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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 Erdös and Spencer for the unweighted case.

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We prove that both multiple Rademacher system and Rademacher chaos possess the property of random unconditional convergence in the space L_∞. This fact combined with some intimate connections between L_∞-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.

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The paper addresses: Random unconditional convergence of Rademacher chaos in L_∞ and sharp estimates for discrepancy of weighted graphs and hypergraphs.
We prove that both multiple Rademacher system and Rademacher chaos possess the property of random unconditional convergence in the space L_∞.