On digit frequencies in {\beta}-expansions

TL;DR

This paper investigates the structure and properties of digit frequency sets in {eta}-expansions, revealing their convexity, continuous dependence on {eta}, and complex extreme point configurations.

Contribution

It characterizes the geometric structure of digit frequency sets, showing their convexity, continuity, and detailed extreme point behavior in relation to {eta}.

Findings

DF({eta}) is a compact convex set with countably many extreme points.

DF({eta}) varies continuously with {eta}.

Generic digit frequency sets have infinitely many extreme points.

Abstract

We study the sets DF({\beta}) of digit frequencies of {\beta}-expansions of numbers in [0,1]. We show that 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 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.