Quantitative bounds in a popular polynomial Szemerédi theorem

We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let P₁, , Pₘ ∈ Z[y] be polynomials with distinct degrees, each having zero constant term. Then there exists a constant c = c(P₁,,Pₘ) > 0 such that any subset A ⊂ \1,2,,N\ of density at least (log N)⁻ᶜ contains a nontrivial polynomial progression of the form x, x+P₁(y), , x+Pₘ(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 .