On Kolmogorov's rearrangement problem and Garsia's conjecture

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.