Unique expansions and intersections of Cantor sets

Abstract

Abstract. To each α ∈ (1/3, 1/2) we associate the Cantor setIn this paper we consider the intersection Γ α ∩ (Γ α + t) for any translation t ∈ R. We pay special attention to those t with a unique {−1, 0, 1} α-expansion, and study the setWe prove that there exists a transcendental number α KL ≈ 0.39433 . . . such that: D α is finite for α ∈ (α KL , 1/2), D αKL is infinitely countable, and D α contains an interval for α ∈ (1/3, α KL ). We also prove that D α equals [0, log 2 − log α ] if and only if α ∈ (1/3, 3− √ 5 2 ]. As a consequence of our investigation we prove some results on the possible values of dim H (Γ α ∩ (Γ α + t)) when Γ α ∩ (Γ α + t) is a self-similar set. We also give examples of t with a continuum of {−1, 0, 1} α-expansions for which we can explicitly calculate dim H (Γ α ∩ (Γ α + t)), and for which Γ α ∩ (Γ α + t) is a self-similar set. We also construct α and t for which Γ α ∩ (Γ α + t) contains only transcendental numbers.Our approach makes use of digit frequency arguments and a lexicographic characterisation of those t with a unique {−1, 0, 1} α-expansion.

Difference of Cantor sets and frequencies in Thue--Morse type sequences

“…Proof. We generalize the proof of [4,Lemma 4.2] given for m = 1. Fix an arbitrary sequence (n i ) of natural numbers.…”

“…In their recent paper [4], Baker and Kong studied the size of the intersection sets Γ q,1 ∩ (Γ q,1 + t) where t is a given real number. The purpose of this paper is to extend their results to the more general case of the sets Γ q,m 1 ∩ (Γ q,m 2 + t), where m 1 and m 2 are arbitrary positive integers.…”

Difference of Cantor sets and frequencies in Thue--Morse type sequences

“…Proof. We generalize the proof of [5,Lemma 4.2] given for m = 1. Fix an arbitrary sequence (n i ) of natural numbers.…”

“…In their recent paper [5], Baker and Kong studied the size of the intersection sets Γ q,1 ∩ (Γ q,1 + t) where t is a given real number. The purpose of this paper is to extend their results to the more general case of the sets Γ q,m1 ∩ (Γ q,m2 + t), where m 1 and m 2 are arbitrary positive integers.…”

Intersections of middle-$α$ Cantor sets with a fixed translation

“…When α ∈ (0, 1/3), Zou, Lu and Li [19] determined when the intersection of middle-α Cantor set with its translation is a self-similar set under the condition that the translation has a unique coding. Recently, Baker and the second author [2] proved that for α ∈ (0, 1/3) it is possible that the intersection of middle-α Cantor set with its translation contains only Liouville numbers.…”