Prescribed distinct-digit growth in countable alphabets

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.