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

Ashwin Sah, MIT
Fine Hall 224

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/loglog…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.