Given an infinite set $A$ of positive integers, $\min A:=a_0$. For $x\geq a_0$ define $f(x)=x-a$, where $a\leq x$, $a\in A$ is greatest possible. Then for positive integer $x$ iterations $x$, $f(x)$, $f(f(x))$, $\dots$ finally come to some element of the set $\{0,1,\dots,a_0-1\}$. Denote this final number $F(x)$. For example, if $A$ is the set of primes, $F(x)$ equals either 0 or 1. Do there always exist frequencies $\lim \frac{|F^{-1}(s)\cap [1,N]|}{N}$ for $s=0,1,\dots,a_0-1$? If not, what is criterion of existing such frequencies? Do they exist, say, for $A$=primes?

Let $A$ be a set of positive integers, $\min A:=a_0$. For $x\geq a_0$ define $f(x)=x-a$, where $a\leq x$, $a\in A$ is the maximal possible. Then for a positive integer $x$ the iterations $x$, $f(x)$, $f(f(x))$, $\dots$ finally come to some element of the set $\{0,1,\dots,a_0-1\}$. Denote this final number by $F(x)$. For example, if $A$ is the set of primes, $F(x)$ equals either 0 or 1. Do there always exist frequencies $\lim \frac{|F^{-1}(s)\cap [1,N]|}{N}$ for $s=0,1,\dots,a_0-1$? If not, what is a criterion of existing of such frequencies? Do they exist, say, for $A$=primes?