Algebraic dependence number and cardinality of generating iterated function systems

For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and Máthé found an invariant, called `algebraic dependence number’, by considering its generating IFSs and isometry invariant self-similar measures. We find an intrinsic quantitative characterisation of this number: it is the dimension over $\mathbb{Q}$ of the vector space generated by the logarithms of all the common ratios of infinite geometric sequences in the gap length set, minus 1. With this concept, we present a lower bound on the cardinality of generating IFS (with or without separation conditions) in terms of the gap lengths of a dust-like set. We also establish analogous result for dust-like graph-directed attractors on complete metric spaces. This is a new application of the ratio analysis method and the gap sequence.

💡 Research Summary

This paper investigates the relationship between the algebraic dependence number of a dust‑like self‑similar set and the cardinality of its generating iterated function systems (IFSs). The algebraic dependence number, originally introduced by Elekes, Keleti and Máthé, is defined as the dimension over ℚ of the vector space generated by the logarithms of all contraction ratios of a generating IFS, minus one. Their definition relies on isometry‑invariant self‑similar measures and is therefore indirect.

The authors provide a completely intrinsic and quantitative characterisation of this invariant. They introduce the gap‑length set GL(K) of a compact dust‑like set K, defined via the discontinuities of the function κ(δ) that counts the number of δ‑equivalence classes (connected components under δ‑chains). This definition coincides with the classical “gap sequence” for subsets of ℝ and is purely geometric.

Using the ratio‑analysis method (originally developed in