# Conditional expanding bounds for two-variables functions over prime fields

Abstract : In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $\Delta(\alpha)>0$ such that if $|A|\asymp p^{\alpha}$ then $\max(|A+A|,|A\cdot A|)\gg |A|^{1+\Delta(\alpha)},$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\asymp|B|\asymp p^{\alpha}$ then $\max(|f(A,B)|,|g(A,B)|)\gg |A|^{1+\Delta(\alpha)}$ for some $\Delta(\alpha)>0$.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00868110
Contributor : François Hennecart <>
Submitted on : Tuesday, October 1, 2013 - 10:07:58 AM
Last modification on : Wednesday, December 12, 2018 - 3:31:05 PM

### Identifiers

• HAL Id : hal-00868110, version 1
• ARXIV : 1309.7580

### Citation

Norbert Hegyvári, François Hennecart. Conditional expanding bounds for two-variables functions over prime fields. European Journal of Combinatorics, Elsevier, 2013, 34, pp.1365-1382. ⟨hal-00868110⟩

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