Upper tails for arithmetic progressions revisited
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916391780614144 |
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| author | Harel, Matan Mousset, Frank Samotij, Wojciech |
| author_facet | Harel, Matan Mousset, Frank Samotij, Wojciech |
| contents | 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 $Ω(N^{-2/k}) \le p \ll 1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08383 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper tails for arithmetic progressions revisited Harel, Matan Mousset, Frank Samotij, Wojciech Probability Combinatorics 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 $Ω(N^{-2/k}) \le p \ll 1$ 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. |
| title | Upper tails for arithmetic progressions revisited |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2409.08383 |