Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

  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},
  • B. Bhattacharya, S. Ganguly, +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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