New estimates for exponential sums over multiplicative subgroups and intervals in prime fields

Abstract

Let H be a multiplicative subgroup of F * p of order H > p 1/4 . We show thatwhere e p (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.

Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity

“…We note that both Korobov [22] and Postnikov [32] formulate their bounds only for prime fields, but the proofs extend to arbitrary finite fields without any changes (at the cost of essentially only typographical changes). On the other hand, in the case of prime fields or finite fields of large characteristic, several better bounds of this kind are known [2,3,5,6,8,12,40] but their underlying methods, based on additive combinatorics, do not extend to arbitrary finite fields (and sometimes apply only to special cases).…”

Weil Sums over Small Subgroups

“…e p f x (p−1)/τ = O(p 1/2 ), (1•3) instantly gives a nontrivial result for subgroups of order τ = #G ≥ p 1/2+ε . In the case of linear polynomials, the bound of [23, theorem 2] has started a series of further improvements which goes beyond this limitation on #G, see [2,4,5,7,11,21] and references therein.…”

“…In the case of linear polynomials, the bound of [ 23 , theorem 2] has started a series of further improvements which goes beyond this limitation on , see [ 2 , 4 , 5 , 7 , 11 , 21 ] and references therein.…”