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

For an odd prime p, let [Formula: see text] 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 [Formula: see text], we also obtain new upper bounds of the following double character sum [Formula: see text] and a triple character sum [Formula: see text] with [Formula: see text] and multiplicative subgroups [Formula: see text] of order [Formula: see text] and [Formula: see text], respectively.