Georgia Institute of Technology College of Sciences

How large of a random subset $D \subset \mathbb{F}_p^n$ does one need to almost surely intersect zero sets cut out by at most $s$ polynomials each of degree at most $k$? We determine the sharp threshold for this problem for all fixed $s$ and $k$. A corollary is that there exists a dense subset $A \subset \mathbb{F}_p^n$ free of k-term arithmetic progressions with common difference in a sufficiently small $D$, improving the lower bound for what is known as Szemerédi’s theorem with random differences. Our bound is the first to capture dependence of $|D|$ on $|A|$ in the finite field setting, giving better dependence than what is known in the integers. Based on joint work with Daniel Altman.