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ON ITERATED PRODUCT SETS WITH SHIFTS

Part of: Harmonic analysis in one variable Additive number theory; partitions

Published online by Cambridge University Press:  21 May 2019

Brandon Hanson ,
Oliver Roche-Newton  and
Dmitrii Zhelezov
Brandon Hanson
Affiliation:
University of Georgia, Athens, GA, U.S.A. email [email protected]
Oliver Roche-Newton
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria email [email protected]
Dmitrii Zhelezov
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary email [email protected]
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Abstract

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$
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 $\unicode[STIX]{x1D6EC}$-constants used in harmonic analysis, applied to Dirichlet polynomials.

MSC classification

Primary: 11P70: Inverse problems of additive number theory, including sumsets 42A70: Trigonometric moment problems
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 831 - 850
Copyright
Copyright © University College London 2019 

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