Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth

Let q be a power of a prime p, let ψ q : F q → C be the canonical additive character ψ q ( x ) = exp ( 2 π i Tr F q / F p ( x ) / p ) , let d be an integer with gcd ( d , q − 1 ) = 1 , and consider Weil sums of the form W q , d ( a ) = ∑ x ∈ F q ψ q ( x d + a x ) . We are interested in how many different values W q , d ( a ) attains as a runs through F q ⁎ . We show that if | { W q , d ( a ) : a ∈ F q ⁎ } | = 3 , then all the values in { W q , d ( a ) : a ∈ F q ⁎ } are rational integers and one of these values is 0. This translates into a result on the cross-correlation of a pair of p-ary maximum length linear recursive sequences of period q − 1 , where one sequence is the decimation of the other by d: if the cross-correlation is three-valued, then all the values are in Z and one of them is −1. We then use this to prove the binary case of a conjecture of Helleseth, which states that if q is of the form 2 2 n , then the cross-correlation cannot be three-valued.