A note on an additive problem with powers of a primitive root (Q2501481)

Let \(p\) be a prime number and \(g\) a primitive root modulo \(p\), and consider the set of differences of powers of \(g\), \(\{g^x-g^y \pmod p: 1 \leq x, y \leq N \}\), where \(N\) is a positive integer with \(N < p\). The author studies a problem about the distribution of this set, that to determine conditions on \(N\) and \(p\) such that the set contains all the residue classes modulo \(p\). Odlyzko conjectured that it suffices to take \(N = p^{1/2+\varepsilon}\), for any \(\varepsilon > 0\), for \(p\) greater than some \(p_\varepsilon\). The best known result was given by \textit{M. Z. Garaev} and \textit{K.-L. Kueh} [Int. J. Math. Math. Sci. 2003, No. 50, 3189--3194 (2003; Zbl 1037.11002)], who proved that one can take \(N=10p^{3/4}\). In this paper, the author improves the bound and shows that we can take \(N = 2^{5/4} p^{3/4}\), for any \(p\). He uses a method involving a simple estimate for a double sum of cosines, which is proved in the paper itself.