Year: 2018
Type: article
Abstract: Let Xk denote the number of k-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots , N\}$ where every element is included independently with probability p. We determine the asymptotics of $\log \mathbb{P}\big (X_{k} \ge \big (1+\delta \big ) \mathbb{E} X_{k}\big )$ (also known as the large deviation rate) where p → 0 with $p \ge N^{-c_{k}}$ for some constant ck > 0, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of p, the large deviation rate up to a constant factor.
DOI: 10.1093/imrn/rny022
Authors: Bhaswar B Bhattacharya, Shirshendu Ganguly, Xuancheng Shao, Yufei Zhao
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2370785362