Entropy for some monoids of natural numbers

In abstract number theory and its applications to statistical physics, the concept of entropy plays an important role. Since entropy is equal to the logarithm of the distribution function, studying the entropy behavior of a monoid is equivalent to solving the inverse problem for this monoid.
The paper considers questions about the asymptotics of entropy for some monoids of natural numbers and monoids of natural numbers with a weight function.
First, the problem is solved for two monoids of the geometric progression type.
Secondly, the results obtained with respect to entropy for monoids with an arbitrary exponential sequence of primes of type $q$ are based on the solution of the inverse problem for monoids of this type obtained earlier by the authors.
To solve this problem, we consider two homomorphisms of the main monoid ${M(\mathbb{P}(q))}$ of type $q$ and the distribution problem reduces to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of $C$ logarithmic $\theta$-power density is introduced.
It is shown that any monoid ${M(\mathbb{P}(q))}$ for a sequence of pseudo-simple numbers $\mathbb{P}(q)$ of type $q$ has upper and lower bounds for the element distribution function of the main basic monoid ${M(\mathbb{P}(q))}$ of type $q$. It is shown that if $C$ is a logarithmic $\theta$-power density for the main monoid ${M(\mathbb{P}(q))}$ of the type $q$ exists, then $\theta=\frac{1}{2}$ and for the constant $C$ the inequalities are valid $ \pi\sqrt{\frac{1}{3\ln q}}\le C\le \pi\sqrt{\frac{2}{3\ln q}}. $
The results obtained are similar to those previously obtained by the authors when solving the inverse problem for monoids generated by an arbitrary exponential sequence of primes of type $q$.
For basic monoids ${M(\mathbb{P}(q))}$ of the type $q$, the question remains open about the existence of a $C$ logarithmic $\frac{1}{2}$-power density and the value of the constant $C$.