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arXiv · 1801.07982

On iterated product sets with shifts

Brandon Hanson·Oliver Roche-Newton·Dmitrii Zhelezov
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Abstract

We prove that, for any finite set $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$| \{(a_1+1)(a_2+1) \cdots (a_k+1) : a_i \in A \}| \geq \frac{|A|^k}{(8k^4)^{kK}}. $$ This result is essentially optimal when $K$ is of the order $c\log|A|$, for a sufficiently small constant $c=c(k)$. Our main tool is a multiplicative variant of the $\Lambda$-constants used in harmonic analysis, applied to Dirichlet polynomials.

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Brandon Hanson, Oliver Roche-Newton, Dmitrii Zhelezov. 2018-01-24. On iterated product sets with shifts. https://doi.org/10.1112/s0025579319000081

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