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On some multiplicative properties of large difference sets

Part of: Linear algebraic groups and related topics Sequences and sets Forms and linear algebraic groups Ergodic theory

Published online by Cambridge University Press:  08 September 2023

Ilya D. Shkredov
Ilya D. Shkredov*
Affiliation:
London Institute for Mathematical Sciences, 21 Albemarle Street, London W1S 4BS, United Kingdom
*
e-mail: [email protected]
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Abstract

In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

Keywords

Difference setssum-productexponential sumsincidences

MSC classification

Primary: 11B13: Additive bases, including sumsets 11B30: Arithmetic combinatorics; higher degree uniformity 11B50: Sequences (mod $m$) 11B75: Other combinatorial number theory
Secondary: 11E57: Classical groups 20G45: Applications to physics 37A05: Measure-preserving transformations
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 5 , October 2024 , pp. 1538 - 1555
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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