Small digitwise perturbations of a number make it normal to unrelated bases

Let $r, g\geq 2$ be integers such that $\log g/\log r$ is irrational. We show that under $r$-digitwise random perturbations of an expanded to base~$r$ real number $x$, which are small enough to preserve $r$-digit asymptotic frequency spectrum of $x$, the $g$-adic digits of $x$ tend to have the most chaotic behaviour.

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