On the set \(ax+bg^ x\pmod p\). (Q1609952)

Let \(g\) be a primitive root modulo prime \(p\). This paper approximates the cardinality of the set \[ {\mathcal M}=\{x\in[0,N]\mid g^x \in[0,M] \text{ and } ax+bg^x <t\} \] where all residues are taken between \(0\) and \(p-1\). This approximation is up to the error term \(O(p^{1/2}\log^3p)\). The proof is short and makes efficient use of a lemma of Mordell bounding the complete sum \(\sum_{x}e(mx+ng^x)\) by \(2p^{1/2}(1+\log p)\) if \(m\) and \(n\) are not both 0 modulo \(p\) and of a lemma of the authors expressing \(\sum_xf(x,g^x)\) in terms of the Fourier transform of \(f(x,y)\) and the above complete sums. Optimal corollaries are deduced concerning the number of \(x\) such that \(x>g^x\) -- this occurs half of the time up to \(7p^{1/2}(1+\log p)^3\) exceptions -- and the moments of \(g^x-x\).