Improved bounds for skew corner-free sets

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 $\mathrm{\Omega }(n^{5\mathrm{/}4})$ 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.