The Distribution of Values of Short Hybrid Exponential Sums on Curves over Finite Fields

Abstract

Abstract. Let p be a prime number, X be an absolutely irreducible affine plane curve over Fp, and g, f ∈ Fp(x, y). We study the distribution of the values of the hybrid exponential sums Sn = Xon n ∈ I for some short interval I. We show that under some natural conditions the limiting distribution of the projections of the sum Sn, n ∈ I on any straight line through the origin is Gaussian as p tends to infinity.

The distribution of short character sums

“…We also note that our method furnishes the same bound (as in Theorem 1) for the rate of convergence of the distribution of S χ,H (x)/ √ H to the standard Gaussian in (1•1), in the case where χ is the Legendre symbol modulo q. Moreover, it is possible to obtain an analogous result to Theorem 1 for more general short exponential sums if one combines our approach with the work of Mak and Zaharescu in [4].…”

“…for non-negative integers r, s. Mak and Zaharescu [4] had previously computed the moments of ReS χ,H (x) (and also those of ImS χ,H (x)) and proved that they are close to the moments of a Gaussian. In our case applying their method leads to weaker error terms.…”

“…Now, using (3•1) along with the fact that e i x = 1 4 . The result follows upon inserting this estimate in (3•2).…”

“…Recently in [4], Mak and Zaharescu generalized Davenport and Erdös approach by investigating the distribution of more general short exponential sums. Their results imply that for any non-real character χ modulo q, both ReS χ,H (x) and ImS χ,H (x) have a limiting Gaussian distribution.…”

Gaussian distribution of short sums of trace functions over finite fields

“…• Davenport-Erdős [DE52] and Mak-Zaharescu [MZ11] directly show that the moments of (1•1) are asymptotically Gaussian and apply the method of moments; • Lamzouri [Lam13] first proves that his probabilistic model is accurate as in step (ii) above. He then remarks that the random variable X modeling the values of the Dirichlet characters itself has moments bounded by those of a Gaussian.…”