On digit frequencies in \(\beta\)-expansions (Q2221028)

The beta-expansions of integers, i.e. the expansions of the representation of real numbers in non-integer bases were introduced by \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 8, 477--493 (1957; Zbl 0079.08901)], yet a lot of work has been published until now.\par The digit frequencies of beta-expansions is examined in this paper and it is shown that Lebesgue almost every number has a beta-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many beta-expansions of the same frequency. The three main theorems of the paper are given at the first paragraph, allowing the reader for a holistic view of the results. The second paragraph consists of all the necessary notation and preliminaries, while in the third paragraph we can se the proofs of the main results. A final and very interesting paragraph is concluding the paper with three open questions including possible generalized theorems and equivalent results.