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On sum-product bases

Authors:	Norbert Hegyvári

Institution:	1. Institute of Mathematics, ELTE TTK, E?tv?s University, H-1117 Pázmány st. 1/c, Budapest, Hungary

Abstract:	We investigate additive-multiplicative bases in $\mathbb {F}_{p}$ . Let $A,B\subseteq \mathbb {F}^{}_{p}$ , s>2, and $\lambda_{1},\lambda_{2},\dots ,\lambda_{s}\in \mathbb {F}^{}_{p}$ . It is proved that $\sum_{i=1}^{s}\lambda_{i}AB=\mathbb {F}_{p}$ , provided min {\|B\|^s/2\|A\|^(s−2)/2,\|A\|^s/2\|B\|^(s−2)/2}>p ^s/2. This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.

Keywords:	Thin bases Sum-product bases
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