再论算术级数的上尾数

Matan Harel, Frank Mousset, Wojciech Samotij
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引用次数: 0

摘要

让 $X$ 是前 $N$ 正整数的 $p$ 偏随机子集中包含的 $k$ 期算术级数的数目。对于所有$\Omega(N^{-2/k}) \le p \ll 1$和所有$t \gg\sqrt{Var(X)}$,我们给出了对数上尾概率$\log\Pr(X \ge E[X] +t)$的渐近尖锐估计,仅排除了一些边界情况。我们特别指出,参数 $(p,t)$ 的空间被划分为三个现象学上截然不同的区域,其中上尾概率要么类似于高斯或泊松随机变量的概率,要么自然地由一个小集合的出现概率来描述,而这个小集合几乎包含了所有多余的 $t$ 级数。我们采用了概率论中的各种工具,包括经典的倾斜论证和马氏集中不等式。然而,主要的技术创新是一个组合结果,它为具有丰富算术结构的集合建立了一个更强版本的 "熵稳定性"。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Upper tails for arithmetic progressions revisited
Let $X$ be the number of $k$-term arithmetic progressions contained in the $p$-biased random subset of the first $N$ positive integers. We give asymptotically sharp estimates on the logarithmic upper-tail probability $\log \Pr(X \ge E[X] + t)$ for all $\Omega(N^{-2/k}) \le p \ll 1$ and all $t \gg \sqrt{Var(X)}$, excluding only a few boundary cases. In particular, we show that the space of parameters $(p,t)$ is partitioned into three phenomenologically distinct regions, where the upper-tail probabilities either resemble those of Gaussian or Poisson random variables, or are naturally described by the probability of appearance of a small set that contains nearly all of the excess $t$ progressions. We employ a variety of tools from probability theory, including classical tilting arguments and martingale concentration inequalities. However, the main technical innovation is a combinatorial result that establishes a stronger version of `entropic stability' for sets with rich arithmetic structure.
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