Sum-Product and Character Sums in finite fields

Cornell University - arXiv, 2011

This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Zürich, November, 2010). We state some conjectures on archimedian and p-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide[4], which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is P n itself.

Journal of the London Mathematical Society, 2006

Our first result is a 'sum-product' theorem for subsets A of the finite field Fp, p prime, providing a lower bound on max(|A + A|, |A · A|). The second and main result provides new bounds on exponential sums

We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{{3/2}}}{q^{{1/4}}$.

Comptes Rendus Mathematique, 2006

We establish bounds on exponential sums x∈F q ψ(x n) where q = p m , p prime, and ψ an additive character on F q. They extend the earlier work [BGK] to fields that are not of prime order (m ≥ 2). More precisely, a nontrivial estimate is obtained provided n satisfies gcd (n, q−1 p ν −1) < p −ν q 1−ε for all 1 ≤ ν < m, ν|m, where ε > 0 is arbitrary.

Comptes Rendus Mathematique, 2003

Our first result is a 'sum-product' theorem for subsets A of the finite field F p , p prime, providing a lower bound on max(|A + A|, |A · A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group F

Annals of Combinatorics

Balog and Wooley have recently proved that any subset A of either real numbers or of a prime finite field can be decomposed into two parts U and V, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function f both the additive energies of U and f pVq are small. Our method is based on bounds of character sums which leads to the restriction #A ą q 1{2 where q is the field size. The bound is optimal, up to logarithmic factors, when #A ě q 9{13. Using f pXq " X´1 we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions ab`ac`bc with variables a, b, c running through three arbitrary subsets of a finite field.

Mathematics of Computation, 2015

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime p.

arXiv (Cornell University), 2020

Let Fq be the finite field with q elements, where q is a prime power and, for each integer n ≥ 1, let Fqn be the unique n-degree extension of Fq. The Fq-orders of an element in Fqn and an additive character over Fqn have been extensively used in the proof of existence results over finite fields (e.g., the Primitive Normal Basis Theorem). In this note we provide an interesting relation between these two objects.

Contemporary Mathematics, 2008

We prove that if A ⊂ F q is such that |A| > q 1 2 + 1 2d , then F * q ⊂ dA 2 = A 2 + • • • + A 2 d times, where A 2 = {a • a ′ : a, a ′ ∈ A}, and where F * q denotes the multiplicative group of the finite field F q. In particular, we cover F * q by A 2 + A 2 if |A| > q 3 4. Furthermore, we prove that if |A| ≥ C 1 d size q 1 2 + 1 2(2d−1) , then |dA 2 | ≥ q • C 2 size C 2 size + 1. Thus dA 2 contains a positive proportion of the elements of F q under a considerably weaker size assumption.We use the geometry of F d q , averages over hyper-planes and orthogonality properties of character sums. In particular, we see that using operators that are smoothing on L 2 in the Euclidean setting leads to non-trivial arithmetic consequences in the context of finite fields.

1999

Let χ be a nontrivial multiplicative character of F p n. We obtain the following results related to Davenport-Lewis' paper [DL] and the Paley Graph conjecture. (1). Let ε > 0 be given. If

Acta Arithmetica, 2002

International Mathematics Research Notices, 2010

We establish improved sum-product bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.

Proceedings of the London Mathematical Society, 1973

Journal of Number Theory, 2000

Cohn's problem on character sums (see [6], p. 202) asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let f be a map from a finite field F to the complex plane such that f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0. In this paper we show that if for all a, b ∈ F * , we have (q − 1) α∈F f (bα)f (α + a) = − α∈F f (bα)f (α), then f is a multiplicative character of F. We also prove that if F is a prime field and f is a real valued function on F with f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0, then α∈F f (α)f (α + a) = −1 for all a = 0 if and only if f is the Legendre symbol. These results partially answer Cohn's problem.