Strong Bounds for 3-Progressions
Zander Kelley, Raghu Meka
TL;DR
The paper proves that there exists a β>0 such that any A⊆{1,...,N} with density |A|/N ≥ 2^{-O((log N)^{β})} contains a nontrivial 3-term progression, improving previous bounds based on logarithmic measurements. The authors introduce a robust, analytic framework starting with finite-field cap-set techniques, including spreadness, regularity, and self-regularity, and then transfer these ideas to the integer setting via a density-increment strategy and Freiman-type embeddings. Two key technical pillars drive the results: (I) near-uniformity from spreadness, enabling strong upper bounds on solution counts, and (II) strong two-sided bounds from self-regularity, yielding near-uniform convolution and stable lower bounds. Combined with a density-increment mechanism and a robust sunflower-type structural lemma, these tools yield many 3-progressions in dense sets of integers, bridging finite-field methods with additive-structure analysis in Z. The work deepens understanding of how additive pseudorandomness leads to arithmetic progressions and suggests new avenues for tightening density thresholds in the integers with potential applications in related combinatorial problems and complexity settings.
Abstract
We show that for some constant $β> 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^β)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N)^{1 + c}$ for a constant $c > 0$. Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to apply some analogous variants of these same techniques, suitably adapted for the more complicated setting of integers.