Let Γ β,N be the N -part homogeneous Cantor set with β ∈ (1/(2N − 1), 1/N ).is called a code of t. Let U β,±N be the set of t ∈ [−1, 1] having a unique code, and let S β,±N be the set of t ∈ U β,±N which make the intersection Γ β,N ∩ (Γ β,N + t) a selfsimilar set. We characterize the set U β,±N in a geometrical and algebraical way, and give a sufficient and necessary condition for t ∈ S β,±N . Using techniques from beta-expansions, we show that there is a critical point βc ∈ (1/(2N − 1), 1/N ), which is a transcendental number, such that U β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), βc), and contains countably infinite many elements if β ∈ (βc, 1/N ). Moreover, there exists a second critical point αc = N + 1 − (N − 1)(N + 3) /2 ∈ (1/ (2N − 1), βc) such that S β,±N has positive Hausdorff dimension if β ∈ (1/ (2N − 1), αc), and contains countably infinite many elements if β ∈ [αc, 1/N ).

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ABSTRACT:In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case.

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An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and planefilling properties of these curves, giving simple geometric criteria for a complete classification. Finally, we show how the automatic structure of the sequences leads to self-similarity of the curves, which turns the planefilling curves in a scaling limit into fractal tiles. For some of these tiles we give a particularly simple formula for the Hausdorff dimension of their boundary.

An automatic sequence is a letter-to-letter coding of a fixed point of a uniform morphism. More generally, morphic sequences are letter-to-letter codings of fixed points of arbitrary morphisms. There are many examples where an, a priori, morphic sequence with a non-uniform morphism happens to be an automatic sequence. An example is the Lysënok morphism a → aca, b → d, c → b, d → c, the fixed point of which is also a 2-automatic sequence. Such an identification is useful for describing the dynamical systems generated by the fixed point. We give several ways to uncover such hidden automatic sequences, and present many examples. We focus in particular on morphisms associated with Grigorchuk groups.

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closedform expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.