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arXiv · 1309.7580

Conditional expanding bounds for two-variables functions over prime fields

Norbert Hegyvári·François Hennecart
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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 $Δ(α)>0$ such that if $|A|\asymp p^α$ then $$ \max(|A+A|,|A\cdot A|)\gg |A|^{1+Δ(α)}, $$ Our aim is to obtain analogous results for related pairs of two-variable functions $f(x,y)$ and $g(x,y)$: if $|A|\asymp|B|\asymp p^α$ then $$ \max(|f(A,B)|,|g(A,B)|)\gg |A|^{1+Δ(α)} $$ for some $Δ(α)>0$.

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Norbert Hegyvári, François Hennecart. 2013-09-29. Conditional expanding bounds for two-variables functions over prime fields. https://doi.org/10.1016/j.ejc.2013.05.021

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