Prescribed distinct-digit growth in countable alphabets

TL;DR

This paper investigates the growth rate of distinct symbols in digit expansions generated by affine iterated function systems on countable alphabets, revealing phase transitions in the Hausdorff dimensions of exceptional sets based on growth rates.

Contribution

It characterizes the Hausdorff dimensions of sets where the digit count grows at specified rates, uncovering a sharp phase transition depending on the growth rate.

Findings

Dimension collapses to a tail index-dependent value for positive linear growth.

Exceptional sets with sublinear growth retain full Hausdorff dimension.

The dimension laws exhibit a phase transition at the linear growth threshold.

Abstract

The number of distinct symbols appearing in digit expansions generated by full-branch affine countable iterated function systems is studied whose branch weights are regularly varying. The Hausdorff dimensions of the exceptional sets in which the distinct-digit count grows at a positive linear rate or at a prescribed sublinear rate are determined. The resulting dimension laws exhibit a sharp phase transition: imposing any positive linear rate forces the dimension to collapse to a value determined solely by the tail index, whereas under a broad class of sublinear growth rates, the exceptional sets retain full Hausdorff dimension.