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arXiv · 2609.18491

On Kolmogorov's rearrangement problem and Garsia's conjecture

Mark Lewko
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Abstract

We give negative answers to Kolmogorov's rearrangement problem and Garsia's conjecture. We construct a complete uniformly bounded orthonormal system for which every rearrangement admits a square-summable series divergent almost everywhere. The construction is built from two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which finds a prescribed permutation pattern as a subsequence of at least one of two longer permutations. Its proof uses Szemerédi's theorem and a counting argument.

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Mark Lewko. 2026-09-16. On Kolmogorov's rearrangement problem and Garsia's conjecture. https://arxiv.org/abs/2609.18491

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