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

For an odd prime p, let 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 ∈ p * , we also obtain new upper bounds of the following double character sum T a , b , c (χ , ℋ 1 , ℋ 2) = ∑ h 1 ∈ ℋ 1 ∑ h 2 ∈ ℋ 2 χ (a + b h 1 + c h 2) and a triple character sum S χ (a , b , ℋ 1 , ℋ 2 ,) = ∑ x ∈ ∑ h 1 ∈ ℋ 1 ∑ h 2 ∈ ℋ 2 χ (x + a h 1 + b h 2) with = { 1 , ... , N } and multiplicative subgroups ℋ 1 , ℋ 2 ⊆ p * of order H 1 and H 2 , respectively. [ABSTRACT FROM AUTHOR]