#### Vol. 8, No. 1, 2019

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Some remarks on the asymmetric sum-product phenomenon

### Ilya D. Shkredov

Vol. 8 (2019), No. 1, 15–41
DOI: 10.2140/moscow.2019.8.15
##### Abstract

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $max\left\{|AB|,|A+C|\right\}$ and $max\left\{|AB|,|\left(A+\alpha \right)C|\right\}$, where $\alpha \ne 0$, in the regime when the sizes of the finite subsets $A,B,C$ of a field differ significantly.

##### Keywords
sum-product, expanders, exponential sums
##### Mathematical Subject Classification 2010
Primary: 11B30, 11P70
##### Milestones
Received: 1 December 2017
Published: 11 August 2018
##### Authors
 Ilya D. Shkredov Steklov Mathematical Institute ul. Gubkina, 9 Moscow Russia 119991 IITP RAS Bolshoy Karetny per. 19 Moscow Russia 127994 MIPT Institutskii per. 9 Dolgoprudnii Russia 141701