Distribution of special sequences modulo a large prime
Let \(p\) be a large prime, \(g\) a primitive root (mod \(p\)), \(N>p\) a given positive integer, \[ A=\{g^x-g^y \text{ (mod \(p\)) }: 1\leq x,y\leq N \} \] and \[ B=\{xy \text{ (mod \(p\)) }: 1\leq x,y\leq N \}. \] The authors show, by elementary estimates, that \[ \# \{h \text{ (mod \(p\)) }: h\notin A \} \ll \frac{p^3}{N^3}, \tag{1} \] \[ \# \{h \text{ (mod \(p\)) }: h\notin B \} \ll \frac{p^3\log p}{N^3}. \tag{2} \] These results are in connection with the following questions: for which values of \(N\) do the sets \(A\), respectively \(B\), contain all residue classes (mod \(p\))? Estimate (1) improves a result of \textit{M. Vâjâitu} and \textit{A. Zaharescu} [Int. J. Math. Math. Sci. 29, 325--331 (2002; Zbl 1010.11003)].
scientific article; zbMATH DE number 2012610