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Double Character Sums over Subgroups and Intervals

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  15 May 2014

MEI-CHU CHANG  and
IGOR E. SHPARLINSKI
MEI-CHU CHANG
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA email [email protected]
IGOR E. SHPARLINSKI*
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
*
✉[email protected]
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Abstract

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We estimate double sums

$$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$
with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums
$$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$
 and give an application to primitive roots modulo $p$ with three nonzero binary digits.

Keywords

character sumsintervalsmultiplicative subgroups of finite fields

MSC classification

Primary: 11L40: Estimates on character sums
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 376 - 390
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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