Agafonov’s theorem for finite and infinite alphabets and probability distributions different from equidistribution
Thomas Seiller, Jakob Grue Simonsen · Ergodic Theory and Dynamical Systems · 2025
Abstract An infinite sequence $\alpha $ over an alphabet $\Sigma $ is $\mu $ -distributed with respect to a probability map $\mu $ if, for every finite string w , the limiting frequency of w in $\alpha $ exists and equals $\mu (w)$ . We prove the following result for any finite or countably infinite alphabet $\Sigma $ : every finite-state selector over $\Sigma $ selects a $\mu $ -distributed sequence from every $\mu $ -distributed sequence if and only if $\mu $ is induced by a Bernoulli distribution on $\Sigma $ , that is, a probability distribution on the alphabet extended to words by taking the product. The primary—and remarkable—consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $\mu $ -distributedness. As a consequence, the shift-invariant measures $\mu $ on $\Sigma ^{\omega }$ , such that any finite-state selector preserves the property of genericity for $\mu $ , are exactly the positive Bernoulli measures.