Double and triple character sums and gaps between the elements of subgroups of finite fields

Abstract

For an odd prime $p$, let ${𝔽}_p$ be the finite field of $p$ elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any $a,b,c\in {𝔽}_p^{\ast }$, we also obtain new upper bounds of the following double character sum $$T_{a,b,c}(\chi ,{{\mathscr{H}}}_1,{{\mathscr{H}}}_2)=\sum _{h_1\in {{\mathscr{H}}}_1}\sum _{h_2\in {{\mathscr{H}}}_2}\chi (a+b{h}_1+c{h}_2)$$ and a triple character sum $$S_{\chi }(a,b,{{\mathscr{H}}}_1,{{\mathscr{H}}}_2,{𝒩})=\sum _{x\in {𝒩}}\sum _{h_1\in {{\mathscr{H}}}_1}\sum _{h_2\in {{\mathscr{H}}}_2}\chi (x+a{h}_1+b{h}_2)$$ with ${𝒩}=\{1,\dots ,N\}$ and multiplicative subgroups ${{\mathscr{H}}}_1,{{\mathscr{H}}}_2\subseteq {𝔽}_p^{\ast }$ of order $H_1$ and $H_2$, respectively.