Quasi-Periodic beta-Expansions and Cut Languages

Abstract

Motivated by the analysis of neural net models between integer and rational weights, we introduce a so-called cut language over a real digit alphabet, which contains finite beta-expansions (i.e. base-beta representations) of the numbers less than a given threshold. We say that an infinite beta-expansion is eventually quasi-periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits, contains an infinite constant subsequence. We prove that a cut language is regular iff its threshold is a quasi-periodic number whose all beta-expansions are eventually quasi-periodic, by showing that altogether they have a finite number of tail values. For algebraic bases beta, we prove that there is an eventually quasi-periodic beta-expansion with an infinite number of tail values iff there is a conjugate of beta on the unit circle. For transcendental beta combined with algebraic digits, a beta-expansion is eventually quasi-periodic iff it has a finite number of tail values. For a Pisot base beta and digits from the smallest field extension Q(beta) over rational numbers including beta, we show that any number from Q(beta) is quasi-periodic. In addition, we achieve a dichotomy that a cut language is either regular or non-context-free and we show that any cut language with rational parameters is context-sensitive.