Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

@article{Bhattacharya2020UpperTL,
title={Upper Tail Large Deviations for Arithmetic Progressions in a Random Set},
author={B. Bhattacharya and S. Ganguly and X. Shao and Y. Zhao},
journal={International Mathematics Research Notices},
year={2020},
volume={2020},
pages={167-213}
}
• B. Bhattacharya, +1 author Y. Zhao
• Published 2020
• Mathematics
• International Mathematics Research Notices
• Let $X_k$ 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 $\log \mathbb{P}(X_k \ge (1+\delta) \mathbb{E} X_k)$ (also known as the large deviation rate) where $p \to 0$ with $p \ge N^{-c_k}$ for some constant $c_k > 0$, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation… CONTINUE READING
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