关于逆Littlewood猜想的注记
Remarks on the inverse Littlewood conjecture
AI总结：
研究在逆Littlewood猜想条件下，集合A的结构特征，证明了A中存在子集具有特定加法性质，并推导出任意k≥3时A中存在长度为k的等差数列。
摘要：
Littlewood猜想由Konyagin和McGehee-Pigno-Smith于1980年代证明，指出若A是整数集合且|A|=N，则‖1_A^ˆ‖_1≥c log N，其中c为绝对常数。本文探讨当‖1_A^ˆ‖_1≤K log N时A必须具备的结构。在该假设下，证明了A中存在一个子集A'，其大小≥N^{0.99}且|A'+A'|≪K^{O(1)}|A'|。由此推导，对于任意k≥3，若N足够大，则A中必须包含长度为k的等差数列。本文分析还改进了常数c的界。
英文摘要：
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
