Character Sums in Finite Fields

Let F q be a finite field of order q with q = p n , where p is a prime. A multiplicative character χ is a homomorphism from the multiplicative group F * q , ·· to the unit circle. In this note we will mostly give a survey of work on bounds for the character sum x χ(x) over a subset of F q. In Section 5 we give a nontrivial estimate of character sums over subspaces of finite fields. §1. Burgess' method and the prime field case. For a prime field F p and when the subset is an interval, Polya and Vinogradov (Theorem 12.5 in [IK]) had the following estimate. Theorem 1.1. (Polya-Vinogradov) Let χ be a non-principal Dirichlet character modulo p. Then a+b m=a+1 χ(m) < Cp 1 2 (log p). 1 2 (log p). Forty four years later Burgess [B1] made the following improvement. Theorem 1.2. (Burgess) Let χ be a non-principal Dirichlet character modulo p. For any ε > 0, there exists δ > 0 such that if b > p 1 4 +ε , then a+b m=a+1 χ(m) p −δ b. Applying the theorem to a quadratic character, one has the following corollary. (The power of 1/ √ e is gained by sieving.) Corollary 1.3. The smallest quadratic non-residue modulo p is at most p 1 4 √ e +ε for ε > 0 and p > c(ε). Note that we always assume ε > 0 and p > c(ε). The proof of the Burgess theorem is based on an amplification argument (due to Vinogradov), a bound on the multiplicative energy of two intervals (Lemma 1.4) and Weil's estimate (Theorem 1.5). The multiplicative energy E(A, B) of two sets A and B is a measure of the amount of common multiplicative structure between A and B. E(A, B) = (a 1 , a 2 , b 1 , b 2) ∈ A × A × B × B : a 1 b 1 = a 2 b 2. Similarly, we can define the multiplicative energy of multiple sets.