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 is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bourgain, J.: Mordell’s exponential sum estimate revisited. J. Am. Math. Soc. 18(2), 477–499 (2005)
Glibichuk, A.A., Konyagin, S.V.: Additive properties of product sets. In: fields of prime order. CRM Proc. Lect. Notes, pp. 1–8. AMS, Providence (2006)
Hart, D., Iosevich, A.: Sums and products in finite fields: an integral geometric viewpoint. arXiv:0705.4256v4 (math. NT)
Konyagin, S.: A sum-product estimate in fields of prime order. arXiv:math.NT/0304217 v1
Tao, T., Vu, V.: Additive Combinatorics. Cambridge Univ. Press, Cambridge (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.
Rights and permissions
About this article
Cite this article
Hegyvári, N. On sum-product bases. Ramanujan J 19, 1–8 (2009). https://doi.org/10.1007/s11139-008-9154-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-008-9154-9