Double Character Sums over Subgroups and Intervals

Abstract

Abstract. We estimate double sumswith a multiplicative character χ modulo p where I = {1, . . . , H} and G is a subgroup of order T of the multiplicative group of the finite field of p elements. A nontrivial upper bound on S χ (a, I, G) can be derived from the Burgess bound if H ≥ p 1/4+ε and from some standard elementary arguments if T ≥ p 1/2+ε , where ε > 0 is arbitrary. We obtain a nontrivial estimate in a wider range of parameters H and T . We also estimate double sumsand give an application to primitive roots modulo p with 3 non-zero binary digits.

On Bilinear Exponential and Character Sums With Reciprocals of Polynomials

“…2) follows. The rest of the proof is identical to that of[8, Lemma 2.9]. ⊓ ⊔ Character sums with uniformly distributed sequences.…”

“…Our results are closely related to various questions about congruences with reciprocals and multiplicative congruences with polynomials of the types considered in [4] and [6,7,9,10,11,18], respectively. We also use one of the results from [8] which we extend to more generic settings; we hope this may find further applications. 1.3.…”

Multiplicative energy of polynomial images of intervals modulo $q$

“…x∈I χ(f (x)) as discussed in [4]. While incomplete character sums over an interval are being studied thoroughly, the notion of character sums with polynomial arguments can be improved with results such as these.…”

Double Character Sums with Intervals and Arbitrary Sets

“…Here we give one of such examples, namely when S is a multiplicative subgroup G ⊆ F p for which the sums W χ (I, S; α, β) have been considered in [10,28] (in the case of constant weights α and β). To simplify the exposition, and enable us to apply a result of Cilleruelo and Garaev [11,Theorem 1] we always assume that #G ≤ p 2/5 .…”

“…To simplify the exposition, and enable us to apply a result of Cilleruelo and Garaev [11,Theorem 1] we always assume that #G ≤ p 2/5 . Note that for large subgroups one can use the results and methods of [10,28]. there exists some δ > 0 such that for any interval I = [1, X] of length X = p ζ+o (1) and a multiplicative subgroup G ⊆ F p of order T = p ξ+o (1) , and complex weights α = {α s } s∈G and β = {β x } x∈I satisfying (1.1), we have…”