Improved bounds for skew corner-free sets

TL;DR

This work advances the understanding of skew corner-free subsets of $[n]^2$ by establishing a Behrend-type lower bound of $s(n) \ge \frac{n^2}{2^{(c_1+o(1))\sqrt{\log_2 n}}}$ and an upper bound of $s(n)=O\left(\frac{n^2}{(\log n)^{c_2}}\right)$, improving the prior $Ω(n^{5/4})$ and $O(n^2(\log\log n)^{-1/73})$ benchmarks. The lower bound uses a high-dimensional Behrend-style construction with a sphere- and inner-product-based fiber analysis and a Freiman-type embedding to $[n]^2$, while the upper bound adapts a Fourier-analytic density-increment framework akin to Roth–Szemerédi methods, including horizontal/vertical $\mathbb{L}^2$-increments and a generalized von Neumann theorem. The combination yields a near-optimal density control for skew corner-free sets and highlights methodological parallels with progress on three-term arithmetic progressions, pointing to potential extensions via modern uniformity techniques. The results contribute to a deeper comprehension of skew corner configurations and inspire open questions about further tightening bounds and leveraging new uniformity methods.

Abstract

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $Ω(n^{5/4})$ obtained by Pohoata and Zakharov. In the other direction, we prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c > 0$. This improves on the previously best known upper bound, coming from Shkredov's work on the corners theorem.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1: Generalised von Neumann theorem
• Proposition 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Proposition 3.6: Horizontal $\mathbb{L}^2$ density increment
• Proposition 3.7: Vertical $\mathbb{L}^2$ density increment
• Lemma 3.8: Dirichlet's approximation theorem