ON DIGIT FREQUENCIES IN β-EXPANSIONS
ON DIGIT FREQUENCIES IN β-EXPANSIONS
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发表时间：
2014
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通讯作者：
P. Boyland;André DE Carvalho;Toby Hall
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作者：
P. Boyland;André DE Carvalho;Toby Hall
研究了[0,1]中数的β-展开式的数字频率集DF（β）。证明了DF（β）是一个紧致凸集，它具有可数个极值点，并随β连续变化；存在一个非平凡闭区间的全测度集合，每个闭区间上DF（β）模都锁定在一个具有有理顶点的常数多面体上；一般数字频率集有无穷多个极值点，这些极值点的各分量是理性独立的。
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.