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arXiv · 2302.05537

Strong Bounds for 3-Progressions

Zander Kelley·Raghu Meka
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Abstract

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ 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.

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Zander Kelley, Raghu Meka. 2023-02-10. Strong Bounds for 3-Progressions. https://arxiv.org/abs/2302.05537

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