On a Problem of H. Cohn for Character Sums

Abstract

Cohn's problem on character sums (see [4], 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∈Ff(bα)f(α+a)=−∑α∈Ff(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∈Ff(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.