AI 中文总结
本文通过构造一致有界正交系统，否定了Kolmogorov重排问题与Garsia猜想，证明存在系统使所有重排均有平方可和级数几乎处处发散，方法基于三角系统副本与组合引理。
AI 中文摘要
我们对Kolmogorov的重排问题和Garsia猜想给出了否定答案。我们构造了一个完备的一致有界标准正交系统，对于该系统，每一个重排都允许一个平方可和级数几乎处处发散。该构造由三角系统的两个副本以不同顺序构建而成。主要成分是一个组合引理，该引理在两个较长的排列中至少一个排列的子序列中找到指定的排列模式。其证明使用了Szemerédi定理和计数论证。
英文摘要
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 $\{\pm1\}$-valued systems with $L^2$ maximal norm at least $c\log\log N$ in every ordering, matching Bourgain's upper bound.
Comments14 pages, no figures. v2: revised exposition and an optimal quantitate construction has been added
