Cyclotomic numbers and a conjecture of Snapper (Q1121943)

In J. Algebra 97, 267-277 (1985; Zbl 0573.12007), \textit{E. Snapper} gave a conjecture equivalent to the following: the cyclotomic numbers \(A_{ij}=Card\{\nu \in {\mathbb{F}}_ q, \chi (\nu)=\zeta^ i, \chi (\nu +1)=\zeta^ j\}\) satisfy \(A_{ij}\to \infty\) as \(q\to \infty\) (for a fixed divisor \(e>1\) of q-1, \(\zeta =\exp (2i\pi /e)\), \(\chi:{\mathbb{F}}_ q^{\times}\to <\zeta>\) of order \(e\)). The proof uses elementary majorization of Jacobi sums \(J(\chi^ i,\chi^ j)\), via the relation \(A_{ab}=e^{-2}\sum_{i,j}\zeta^{- (ai+bj)}J(\chi^ i,\chi^ j) \).