Restricted sums in a field

2007

This is the origin of paper ‘On a Question of Davenport and Lewis on Character Sums and Primitive Roots in Finite Fields’. There is still a little to be typed. Abstract Let A ⊂ Fp with |A| > p and |A + A| < C0|A|. We give explicit constants k = k(C0, ε) and κ = κ(C0, ε) such that |Ak| > κp. The tools we use are Garaev’s sum-product estimate, Freiman’s Theorem and a variant of Burgess’ method. As a by-product, we also obtain similar result for proper generalized progression in Fp.

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

Acta Arithmetica

Acta Arithmetica, 2001

ArXiv, 2017

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. If $q^h-1$ has a prime divisor $r$ with $r=O((h\log q)^c)$, then there is a constant $0 0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+\epsilon'}(q^h))$ containing a non-quadratic element in $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $O(\sqrt{q})$ containing a non-quadratic element in $\mathbb{F}_{q^2}$, essentially improving a $O(q)$ construction by Coulter and Kosick \cite{CK}. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $\phi(q^h-1)$ is very small, where $\phi$ is the Euler's totient function.

Finite Fields and Their Applications, 2008

The subset sum problem over finite fields is a well known NPcomplete problem. It arises naturally from decoding generalized Reed-Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematicial point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we get some information on the decoding problem of Reed-Solomon codes.

Duke Mathematical Journal, 2008

Let χ be a nontrivial multiplicative character of F p n. We obtain the following results. (1). Let ε > 0 be given. If B = { P n j=1 x j ω j : x j ∈ [

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.

Finite Fields and Their Applications, 2022

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.

Journal d'Analyse Mathématique, 2009