On digit frequencies in 𝛽-expansions

Boyland, Philip; de Carvalho, André; Hall, Toby

doi:10.1090/tran/6617

On digit frequencies in $\beta$-expansions
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by Philip Boyland, André de Carvalho and Toby Hall PDF

Trans. Amer. Math. Soc. 368 (2016), 8633-8674 Request permission

Abstract:

We study the sets $\operatorname {DF}(\beta )$ of digit frequencies of $\beta$-expansions of numbers in $[0,1]$. We show that $\operatorname {DF}(\beta )$ is a compact convex set with countably many extreme points which varies continuously with $\beta$; that there is a full measure collection of non-trivial closed intervals on each of which $\operatorname {DF}(\beta )$ mode locks to a constant polytope with rational vertices; and that the generic digit frequency set has infinitely many extreme points, accumulating on a single non-rational extreme point whose components are rationally independent.

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