Double Character Sums over Subgroups and Intervals

Abstract

We estimate double sums $$S_{\chi }(a,I,G)=\sum _{x\in I}\sum _{\lambda \in G}\chi (x+a\lambda ),1\le a<p-1,$$ with a multiplicative character $\chi$ modulo $p$ where $I=\{1,\dots ,H\}$ and $G$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{\chi }(a,I,G)$ can be derived from the Burgess bound if $H\ge p^{1/4+\epsilon }$ and from some standard elementary arguments if $T\ge p^{1/2+\epsilon }$, where $\epsilon >0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$T_{\chi }(a,G)=\sum _{\lambda ,\mu \in G}\chi (a+\lambda +\mu ),1\le a<p-1,$$ and give an application to primitive roots modulo $p$ with $3$ non-zero binary digits.

References in corpus (3)

• Some new inequalities in additive combinatorics
• Incomplete exponential sums over exponential functions
• On exponential sums over multiplicative subgroups of medium size

Cited by in corpus (2)

• Double Character Sums with Intervals and Arbitrary Sets in Finite Fields
• On Bilinear Exponential and Character Sums with Reciprocals of Polynomials