Bounds of double multiplicative character sums and gaps between residues of exponential functions

We derive two new upper bounds on the double multiplicative character sum over subgroups and intervals. R_χ(a,g,I,N)=∑_x=1 ^H|∑_n=1 ^Nχ(x+agⁿ)| where χ is a multiplicative character modulo a prime p, H and N are positive integers and a and g are integers with gcd. (ag, p) = 1. One bound is unconditional and based on a recent result of Cilleruelo and Garaev (2014), the other bound is conditional on the Generalised Riemann Hypothesis (GRH). These bounds complement and improve in some ranges on the recent results of Chang and Shparlinski (2014).