Effective bounds for Roth's theorem with shifted square common difference

Additional Talks and Lectures
Monday, November 13, 2023 - 4:00pm for 1.5 hours (actually 80 minutes)
Skiles 005
Ashwin Sah – MIT
Ernie Croot and Cosmin Pohoata

Let $S$ be a subset of $1 ,…, N$ avoiding the nontrivial progressions $x, x+y^2-1, x+2(y^2-1)$. We prove that $|S| < N/\log \log \cdots \log(N)$, where we have a fixed constant number of logarithms. This answers a question of Green, and is the first effective polynomial Szemerédi result over the integers where the polynomials involved are not homogeneous of the same degree and the underlying pattern has linear complexity. Joint work with Sarah Peluse and Mehtaab Sawhney.