Linear equations over \(\mathbb F_ p\) and moments of exponential sums.
The author obtains an \(\mathbb{F}_p\) analogue of the rearrangement theorems due to Gabriel, Hardy and Littlewood in the version given by the author [J. Comb. Theory, Ser. A 83, 251--267 (1998; Zbl 0907.11009)]. The original ``integral'' version follows from the \(\mathbb{F}_p\) version. He also obtains new estimates, a set \(A\) of residues modulo \(p\) being given, on the number of \(x\in \mathbb{Z}/p\mathbb{Z}\) such that \(\sum_{a\in A}\exp(2\pi iax/p)\) is ``large''. The result obtained can be seen as an \(\mathbb{F}_p\) analogue of a result by \textit{A. A. Yudin} [Teoretiko-Chisl. Issled. Spektru Markov. Struktur. Teor. Slozhen. Mnozhestv, 163--161 (1973; Zbl 0321.10047)].
scientific article; zbMATH DE number 1820801