On digit frequencies in β-expansions

Abstract

We study the sets DF(β) of digit frequencies of β-expansions of numbers in [0,1]. We show that DF(β) is a compact convex set with countably many extreme points which varies continuously with β; that there is a full measure collection of non-trivial closed intervals on each of which DF(β) 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.

39 pages, 10 figures. Modifications from version 1: digit frequency set now proved compact rather than having closure in its definition. Sections 4.3 and 4.4 are new