Large Deviations for Arithmetic Progressions

on YouTube ↗59m

" summary

Proves the sharp upper-tail asymptotics for k-term AP counts when p does not decay too fast, solving the variational problem that gives an explicit constant and supplying a nonlinear large-deviation principle down to p ≥ N^{-α} with α = 1/18 for k=3 and roughly α ≈ 1/(6k^2) in general. Methods combine nonlinear LDPs (Chatterjee–Dembo style), Gowers norms and inverse theorems, Fourier for 3-AP, and a Chinese‑remainder sampling device to compress gradient complexity; conjectures the relevant Gaussian width ~ √n and records current quantitative gaps.

AI-written summary · about

" chapters (10)

1. 0:00Large deviations intro
2. 3:45Core problem and graph analogy
3. 8:44Upper tail probabilities
4. 13:59Nonlinear large deviation principles
5. 22:27Constants in bounds
6. 25:18Arithmetic progressions focus
7. 35:04Gaussian width conjectures
8. 44:18Fourier analysis approach
9. 50:37Sampling and Chinese Remainder Theorem techniques
10. 56:36Future directions in higher-order