TY - JOUR

T1 - On iterated product sets with shifts, ii

AU - Hanson, Brandon

AU - Roche-Newton, Oliver

AU - Zhelezov, Dmitrii

N1 - Funding Information:
Oliver Roche-Newton was partially supported by the Austrian Science Fund FWF Project P 30405-N32. Dmitrii Zhelezov was supported by the Knut and Alice Wallenberg Foundation Program for Mathematics 2017.
Publisher Copyright:
© 2020, Mathematical Science Publishers. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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 · · · ak: a1, …, ak ∈ 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 KC |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.

AB - 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 · · · ak: a1, …, ak ∈ 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 KC |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.

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U2 - 10.2140/ant.2020.14.2239

DO - 10.2140/ant.2020.14.2239

M3 - Article

AN - SCOPUS:85091527982

VL - 14

SP - 2239

EP - 2260

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 8

ER -