Gap sequences of McMullen sets

Abstract

We study the gap sequence of totally disconnected McMullen sets. Our result shows that if every horizontal line in the McMullen set is nonempty, then the gap sequence is unrelated to the box dimension. This implies that in such situations, the separation properties of McMullen sets are quite different from that of self-similar sets.

Gap sequences and Topological properties of Bedford-McMullen sets

“…Next theorem reveals that whether the Bedford-McMullen set satisfies the CSC or not depends on the topological properties of E. The theorem also generalizes the main result of [22], where the Bedford-McMullen sets are required to be totally disconnected. The proof is in Section 4.…”

“…Next lemma gives the second part of Theorem 2.1, and the upper bound estimate of n E (N −k ) in the proof is similar as in [22]. Proof.…”

“…In order to prove the second part of Theorem 2.1, we need the following fact obtained in [22]. We add a proof here for the convenience of readers.…”

“…In [22], Miao, Xi and Xiong obtained the asymptotic estimate of gap sequences of totally disconnected Bedford-McMullen sets. In [20], Liang and Ruan studied the gap sequences of fractal squares, where all (possible) nontrivial connected components are parallel line segments.…”

“…Note that fractal squares can be regarded as a special class of Bedford-McMullen sets, see Subsection 2.1 for definitions. Inspired by [20,22], we reveal the connection between topological properties and gap sequences and obtain the asymptotic estimate of gap sequences of Bedford-McMullen sets without any artificial restriction. For each w = (w (1) , w (2) ) ∈ D, we define a self-affine transformation on R 2 by (2.1)…”