On the behavior of the spectrum of the limit frequencies of digits under perturbations of a real number (Q845219)

Let \(\Lambda_r(t)\) denote the set of all limit points (in the topology of weak convergence) of the sequence of empirical measures \((1/N)\sum_{k=0}^{N-1}\delta(t r^k\mod 1)\) (for a positive integer \(N\)) on \(T:=[0,1)\), where \(\delta(x)\) is the probability measure concentrated at the point \(x\). Denote \(\lambda\) by the Lebesgue measure on \(T\) belonging to the set of probability measures on Borel subsets \(T\) invariant with respect to the \(r\)-adic transformation \(T\to T\) defined by the formula \(t\to r t\pmod 1\) for any positive integer~\(r\). In this paper, the author proves that if \(x\) is obtained as a result of the action of decaying (not too rapidly) random statistically independent digitwise perturbations on the number \(t\), written in the \(r\)-adic number system, then, almost surely, relations (a) \(\Lambda_g(x)=\Lambda_g(t)\) for all \(g\sim r\); and (b) \(\Lambda_g(x)=\{\lambda\}\) for all \(g\not\sim r\); hold.