On the distribution of elements semigroups of natural numbers II

Suppose there is subset $A$ of positive integers from the interval $[1,q]$ with the following condition. If the elements $a,b$ of $A$ and $ab$ is at most $q$, then $ab$ belongs to $A$. In additition let also know that $|A|<q^{\nu}$, $\nu$ — is some fixed number, not exceeding 1. In this paper we consider the question of the number of elements belonging to $A$ on the interval with length substantially less than $q$, — on the interval $[1, x]$, where $x$ is much smaller than an arbitrary power of $q$.
In this task, in the case when $A$ — is a special set and with certain restrictions on $|A|$ and $x$, there exists some results. So, from the work of J. Bourgain, S. Konyagin and I. Shparlinskii there are nontrivial estimates in the case when $A$ — a multiplicative subgroup of invertible elements of the residue ring modulo prime.
The initial problem generalize it to the case of semigroups instead of multiplicative subgroups. It should be noted that there are quite definite results on this task. The main result of this work is to derived a new estimate on the number of elements of the semigroup of natural numbers given short interval from 1 to $x$. These estimates are meaningful when $x$ is much smaller than any power of $q$. More precisely, let $A$ — our semigroup, $g: =\frac{\log{\log x}}{\log{\log q}}, x = q^{o (1)}$ for $q$ tends to infinity. Then the number of elements of $A$ in the interval $(1,x)$ does not exceed $x^{1-C (g,\nu)+o(1)}$, where $C(g,\nu )$ — some clearly written positive function. Previous result relates to the estimation of function $C(g,\nu)$, a new estimate for the $C(g,\nu)$ improves the previous result for a certain range of parameters $(g,\nu)$.
We essentially use in the proof the distribution of smooth numbers, the numbers with a large part of the smooth part, estimates on the number of divisors of a fixed number in a given interval. We use some results of J. Bourgain, S. Konyagin and I. Shparlinski.
Bibliography: 15 titles.