Algebraic dependence numbers and the cardinality of generating iterated function systems

Abstract

For a dust-like self-similar set (generated by an IFS of similarities 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. Using this, we present a lower bound on the minimal cardinality of generating IFSs (with or without separation conditions) in terms of the gap lengths of a dust-like self-similar set. We also establish an analogous result for dust-like graph-directed attractors on complete metric spaces, and present a new proof of the logarithmic commensurability theorem for IFSs with the strong separation condition. These are new applications of the ratio analysis method and the gap sequence.