LOCAL UNIQUENESS OF VECTOR MATRICES RELATED TO EXPONENTIAL SUMS

Abstract

Enflo's approach to estimate exponential sums leads to problems in the interface between number theory, harmonic analysis, and geometry of euclidean space. In particular, he considers the following problem. Let θ be a p-periodic polynomial and assume that for all j, m ≠ 0 mod p. Is then θ(k) = (r/p)k ³ +(s/p)k ² + (t/p)^k + γ? We formulate this problem in terms of symmetric matrices where the entries are vectors and the vectors in each row are pairwise orthogonal, and we show that certain vector matrices are locally unique if and only if p is a prime number. We also show that there are no solutions to Enflo's problem when p = 4 or p = 8.