Prescribed distinct-digit growth in countable alphabets

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.

💡 Research Summary

The paper investigates the growth of the number of distinct symbols (the “distinct‑digit count”) that appear in symbolic expansions generated by full‑branch affine iterated function systems (IFS) on the unit interval with a countable alphabet. The IFS is defined by a probability vector (p_k)_{k∈ℕ} such that the interval is partitioned into disjoint subintervals I_k of length p_k, and each branch maps its subinterval affinely onto the whole interval. Under Lebesgue measure the digit process d_n (the index of the subinterval visited at step n) is independent and identically distributed with P(d_n = k) = p_k. The authors assume that the tail of (p_k) is regularly varying with index ρ ≥ 1, i.e. p_k ≍ k^{‑ρ}L(k) for a slowly varying function L. This setting makes the distinct‑digit count D_n identical to the number of occupied boxes after n draws in an infinite‑urn occupancy scheme with box probabilities p_k. Classical occupancy theory then yields the typical growth law D_n ∼ C·n^{1/ρ} (with an explicit constant involving the Gamma function) and a law‑of‑the‑iterated‑logarithm describing optimal fluctuations.

The main focus is not the typical behaviour but the size (in terms of Hausdorff dimension) of exceptional sets where D_n grows at a prescribed rate. Two families of level sets are defined:

1. Linear growth sets
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