Double exponential sums with exponential functions

Abstract

We obtain several estimates for double rational exponential sums modulo a prime [Formula: see text] with products [Formula: see text] where both [Formula: see text] and [Formula: see text] run through short intervals and [Formula: see text] is fixed integer. We also obtain some new estimates for the number of points on exponential modular curves [Formula: see text] and similar.

Double exponential sums and congruences with intervals and exponential functions modulo a prime

“…Recently, Shparlinski and Yau [10] obtained a number of new explicit estimates on S a,p ( ᾱ, β; N , M). One of the features of [10] is that some of the estimates given there work well for certain ranges of N and M below the critical value p 1/2 .…”

“…Recently, Shparlinski and Yau [10] obtained a number of new explicit estimates on S a,p ( ᾱ, β; N , M). One of the features of [10] is that some of the estimates given there work well for certain ranges of N and M below the critical value p 1/2 . They also noted, that the work of Roche-Newton, Rudnev and Shkredov [9] leads to certain nontrivial bounds in the range M > p 1/3+c , for any positive constant c. Nevertheless, in some very interesting cases (for example, if N, M < p 1/3 ) these estimates become trivial and one naturally asks what can be done in these cases.…”

Bounds for Triple Exponential Sums With Mixed exponential and Linear Terms

“…have been considered by Shparlinski and Yau [8]. For the case where g is not necessarily a primitive root of p, bounds have been established under the condition I = {1} and α m = β n = 1 by Kerr [2], but the method employed there also works for general I as the bounds depend only on the norm.…”

“…8 . Using the same technique as in[5, Lemma 3.14] and the bound in [4, Corollary 19], we obtain the trivial bound…”

Bounds of triple exponential sums with mixed exponential and linear terms

“…Similar double exponential sums has already been considered. In particular, sums of the form S(A, B; I, J ) = m∈I n∈J α m β n e p (amg n ) has been considered in the work by Shparlinski & Yau [7]. For the case when g is not necessary a primitive root of p, bounds has been established under the condition I = {1} and α m = β n = 1 by Kerr [2] but the same method imployed there also works for the general I as the bound depend only on the norm.…”