Distribution of elements of cosets of small subgroups and applications (Q2888915)

Let \(p\) be a prime number, \(\mathbb F_p\) be the field of the prime order, \(G\) be a multiplicative subgroup of \(\mathbb F_p\setminus\{0\}\), and \(a\in\mathbb F_p\) be an arbitrary element. The authors study the numbers NEWLINE\[NEWLINEU(k,G,a)= \{x: x\in aG,\;|x|\leq k\},\quad V(k,G,a)= \{p: x\in aG,\;\| x\|\leq k\},NEWLINE\]NEWLINE where \(|x|\) and \(\| x\|\) are the least integer and rational heights, correspondingly. The results are effective in the case \(\log\,| G|= o(\log p)\), roughly. There are two applications of the bounds found: to the number of fixed points of the discrete logarithm and to the simultaneous distribution of monomials \(x^{k_1}\), \(x^{k_2}\).