On iterated product sets with shifts II

Autor: Dmitrii Zhelezov, Brandon Hanson, Oliver Roche-Newton
Rok vydání: 2018
Předmět:
Zdroj: Algebra Number Theory 14, no. 8 (2020), 2239-2260
DOI: 10.48550/arxiv.1806.01697
Popis: The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$. Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C(\gamma)$ such that for any $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any $c_1,c_2 \in \mathbb Q \setminus \{0\}$, there are at most $K^C|A|^{\gamma}$ solutions to \[ c_1x + c_2y =1 ,\,\,\,\,\,\,\, (x,y) \in A \times A. \] In particular, this result gives a strong bound when $K=|A|^{\epsilon}$, provided that $\epsilon >0$ 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 utilise a query-complexity analogue of the polynomial Freiman-Ruzsa conjecture, due to Zhelezov and P\'alv\"olgyi. This new tool replaces the role of the complicated setup of Bourgain and Chang, which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.
Comment: This paper has shortened considerably, as a consequence of an application of a new result of Zhelezov and P\'alv\"olgyi, see arXiv:2003.04648. This version will appear in Algebra and Number Theory. This paper is a sequel to arXiv:1801.07982, although it can be read independently and does not depend on results from the original paper
Databáze: OpenAIRE
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