PreprintCC BY 4.0arXiv
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 finite construction uses two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which guarantees a prescribed ordering along an arithmetic progression in at least one of two related permutations. Its proof uses Szemerédi's theorem and a counting argument. A Walsh variant gives N-term {±1}-valued systems with L2 maximal norm at least cloglog N in every ordering, matching Bourgain's upper bound.
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BibTeX
@misc{lewko2026kolmogorovs,
title = {{On Kolmogorov's rearrangement problem and Garsia's conjecture}},
author = {Mark Lewko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18491},
eprint = {2609.18491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18491},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Kolmogorov's rearrangement problem and Garsia's conjecture
AU - Mark Lewko
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.18491
UR - https://arxiv.org/abs/2609.18491
AB - 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 finite construction uses two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which guarantees a prescribed ordering along an arithmetic progression in at least one of two related permutations. Its proof uses Szemerédi's theorem and a counting argument. A Walsh variant gives N-term {±1}-valued systems with L^2 maximal norm at least cloglog N in every ordering, matching Bourgain's upper bound.
ER -