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 $\Delta (\alpha )>0$.

References in corpus (2)

• Explicit constructions of extractors and expanders
• Szemeredi-Trotter type theorem and sum-product estimate in finite fields

Cited by in corpus (1)

• Sum-Product Type Estimates over Finite Fields