Quantitative bounds in a popular polynomial Szemerédi theorem

Abstract

We obtain polylogarithmic bounds in the polynomial Szemerédi 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 asymptotically at least as large as in a random set of the same density as $A$.

25 pages. Comments are welcome