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On intersections of Cantor sets: Hausdorff measure. (English) Zbl 1294.11134

Let \(n\geq 3\) be an integer. Let \(D:=\{d_k : k=1,2,\dots,m\}\) with \(m<n\) and \(d_k<d_{k+1}\) be a subset of \(\{0,1,\dots, n-1\}\). The present paper studies the Hausdorff measure of the intersection of the so-called deleted digits Cantor set \(C=C_{n,D}:=\left\{\sum_{k=1}^{\infty} {x_k \over n^k} : x_k\in D \;\forall k\geq 1\right\},\) and its right translate: \(C+t:=\{x+t : x\in C\}\), where \(t\in (0,1)\) is a real number.
The main results of this paper are based on the condition that \(D\) is sparse, i.e., \(|\delta-\delta'|\geq 2\) for all \(\delta\neq \delta'\) in \(\Delta:=D-D=\{d_j-d_k: d_j,d_k \in D\}\).
Let \(t=\sum_{j=1}^{\infty} {t_j \over n^j}\) be the \(n\)-ary representation of \(t\in (0,1)\). Denote by \(\lfloor t\rfloor_k:=\sum_{j=1}^{k} {x_j \over n^j}\) the truncation of \(t\) to the first \(k\) \(n\)-ary places. Let \(C_0=[0,1]\) and for \(k\geq 1\) let \(C_k:=\{\sum_{k=1}^{\infty} {x_k \over n^k} : x_k\in D, 1\leq j \leq k\}\) be the union of \(k\)-th level intervals. Write \(\sharp B\) the number of elements in the finite set \(B\). Define \(\mu_t(0)=1\) and inductively \(\mu_t(k+1)=\mu_t(k) \cdot \sharp (D-t_{k+1})\cap (D\cup(D+1))\) if one of the intervals in \(C_k\) is also an interval in \(C_k +\lfloor t\rfloor_k\); \(\mu_t(k+1)=\mu_t(k) \cdot \sharp (D-n+t_{k+1})\cap (D\cup(D-1))\) if the \({1\over n^k}\)-right translate of one of the intervals in \(C_k\) is an interval in \(C_k +\lfloor t\rfloor_k\); and \(\mu_t(k+1)=0\) otherwise. Let \( \beta_t:=\liminf_{k\to\infty}{\log \mu_t(k) \over k\log m}, \quad \text{and} \quad L_t:=\liminf_{k\to\infty}{\mu_t(k)\over m^{k\beta_t}}. \) The authors prove that if \(D\) is sparse, \(t\in(0,1)\) does not admit a finite \(n\)-ary representation, and \(C\cap (C+t)\) is not empty, then \( m^{-\beta_t}L_t \leq \mathcal{H}^s(C\cap (C+t)) \leq L_t,\) where \(s=\beta_t\log m/\log n\), and \(\mathcal{H}^s\) is the \(s\)-dimensional Hausdorff measure.
The authors also give an overview of previous works on Hausdorff dimension and Hausdorff measure of the set \(C\cap (C+t)\) where some stronger conditions on \(D\) were assumed. In particular, they point out that some similar estimates of Hausdorff measure for a class of Cantor sets (called homogeneous Cantor sets) were obtained by D.-J. Feng et al. [Sci. China, Ser. A 40, No. 5, 475–482 (1997; Zbl 0881.28003)] and C. Q. Qu et al. [Acta Math. Sin., Engl. Ser. 17, No. 1, 15–20 (2001; Zbl 0990.28003)].

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
28A78 Hausdorff and packing measures
28A80 Fractals
37F99 Dynamical systems over complex numbers
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