On iterated product sets with shifts, ii

The main result of this paper is the following: for all b ∈ Z there exists k = k(b) such that max{|A^(k) |, |(A + u)^(k) |} ≥ |A|^b, for any finite A ⊂ Q and any nonzero u ∈ Q. Here, |A^(k) | denotes the k-fold product set {a₁ · · · a_k: a₁, …, a_k ∈ A}. Furthermore, our method of proof also gives the following l_∞ sum-product estimate. For all γ > 0 there exists a constant C = C(γ ) such that for any A ⊂ Q with |AA| ≤ K |A| and any c₁, c₂ ∈ Q \ {0}, there are at most K^C |A|^γ solutions to c₁ x + c₂ y = 1, (x, y) ∈ A × A. In particular, this result gives a strong bound when K = |A|^ɛ, provided that ɛ > 0 is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilize a query-complexity analogue of the polynomial Freiman–Ruzsa conjecture, due to Pälvölgyi and Zhelezov (2020). This new tool replaces the role of the complicated setup of Bourgain and Chang (2004), which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.