On Weil sums, conjectures of Helleseth, and Niho exponents

Let F be a finite field, μ fixed additive character and s an integer coprime to |F×|. For any a∈F, the corresponding Weil sum is defined WF,s(a)=∑x∈Fμ(xs−ax). The spectrum counts distinct values of as runs through invertible elements in field. value these sums size are particular interest, they link problems coding information theory other areas math such number arithmetic geometry. In setting Niho exponents, we examine sum, its bounds spectrum. As consequence, give new proof Vanishing Conjecture Helleseth (1971) on presence zero case exponents. We also state conjecture for when contains at least five elements, prove it certain class sums.