Quantitative bounds in a popular polynomial Szemerédi theorem

Abstract: We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) > 0$ such that any subset $A \subset {1,2,\dots,N}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective ``popular'' version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is almost optimal.