Abstract
Let H be a multiplicative subgroup of Fp⁎ of order H>p1/4. We show thatmax(a,p)=1|∑x∈Hep(ax)|≤H1−31/2880+o(1), where ep(z)=exp(2πiz/p), which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product nx with x∈H and n∈N for a short interval N of consecutive integers.
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