Bounds for sets with no polynomial progressions

Abstract

Let $P_1,\dots ,P_m\in \mathbb{Z}[y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset $A$ of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log N{)}^{c_{P_1,\dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

55 pages; v2: included a new theorem controlling general polynomial progressions by Gowers norms; v3: added to the introduction and stated some intermediate results in greater generality; v4: referee suggestions incorporated