Simultaneously Non-convergent Frequencies of Words in Different Expansions

TL;DR

This paper investigates the Hausdorff dimension of sets defined by word frequency accumulation points in different expanding maps, showing that intersections preserve dimension and that non-convergent frequency sets have full dimension.

Contribution

It proves that the Hausdorff dimension of intersections of sets defined by word frequency accumulation points remains equal to the infimum of their individual dimensions, extending to $eta$-shifts.

Findings

Dimension is preserved under intersection of sets from different expansions.

Sets of numbers with non-convergent frequencies have full Hausdorff dimension.

Results apply to a dense set of $eta$-shifts.

Abstract

We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. It is shown in this text that the dimension is preserved when sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full dimension even after countable intersections. We prove these results also for a dense set of $\beta$-shifts.