On the distribution of elements of semigroups of natural numbers (Q2857932)

Let \(A\) be a multiplicative subgroup of positive integers such that \(|\{n\in A; n\leq q\}|<g^\nu\) for some real \(q\) and \(\nu<1\). The author proves the upper estimates for \(f(x)=|A\cap [1,x]|\) of the following type for \(x=(\log q)^u\):NEWLINENEWLINE (1) if \(\log\log x=o(\log\log q)\) then NEWLINE\[NEWLINE\frac{f(x)}{x}\leq \exp\{-(C+o(1))u(1-\nu^2)\log(u(1-v)^2)\}NEWLINE\]NEWLINE with an absolute constant \(C\);NEWLINENEWLINE (2) if \(\gamma=(\log\log x)/(\log\log q)\) and \(\log x=o(\log q)\) then NEWLINE\[NEWLINEf(x)\leq x^{1-\max\{L_\gamma,C_\gamma\}+o(1)}\quad\text{as}\;q\to\infty,NEWLINE\]NEWLINE with explicitly given constants \(L_\gamma\) and \(C_\gamma\) depending on \(\gamma\) and \(\nu\).