# Besicovitch subsets of self-similar sets

Ji-Hua Ma[1]; Zhi-Ying Wen[2]; Jun Wu[1]

• [1] Wuhan University, Department of Mathematics, Wuhan 430072 (Rép. Pop. Chine)
• [2] Tsinghua University, Department of mathematics, Beijing 10084 (Rép. Pop. Chine)
• Volume: 52, Issue: 4, page 1061-1074
• ISSN: 0373-0956

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## Abstract

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Let $E$ be a self-similar set with similarities ratio ${r}_{j}\left(0\le j\le m-1\right)$ and Hausdorff dimension $s$, let $\stackrel{\to }{p}\left({p}_{0},{p}_{1}\right)...{p}_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$E\left(\stackrel{\to }{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}{\chi }_{j}\left({x}_{k}\right)={p}_{j},\phantom{\rule{1.0em}{0ex}}0\le j\le m-1\right\},$where ${\chi }_{j}$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_{H}\left(E\left(\stackrel{\to }{p}\right)\right)={dim}_{P}\left(E\left(\stackrel{\to }{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_{j}}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_{j}}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\stackrel{\to }{p}=\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then${ℋ}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={ℋ}^{s}\left(E\right),\phantom{\rule{0.277778em}{0ex}}{𝒫}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={𝒫}^{s}\left(E\right),$moreover both of ${ℋ}^{s}\left(E\right)$ and ${𝒫}^{s}\left(E\right)$ are finite positive;(ii) If $\stackrel{\to }{p}$ is a positive probability vector other than $\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then the gauge functions can be partitioned as follows${ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⇔\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha ,$${𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha .$

## How to cite

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Ma, Ji-Hua, Wen, Zhi-Ying, and Wu, Jun. "Besicovitch subsets of self-similar sets." Annales de l’institut Fourier 52.4 (2002): 1061-1074. <http://eudml.org/doc/116003>.

@article{Ma2002,
abstract = {Let $E$ be a self-similar set with similarities ratio $r_j (0\le j\le m-1)$ and Hausdorff dimension $s$, let $\vec\{p\}(p_0,p_1)\ldots p_\{m-1\}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$E(\vec\{p\})=\Big \lbrace x\in E\colon \lim \_\{n\rightarrow \infty \}\{1\over n\} \sum \_\{k=1\}^\{n\}\chi \_\{j\}(x\_k)=p\_j,\quad 0\le j\le m- 1\Big \rbrace ,$where $\chi _j$ is the indicator function of the set $\lbrace j\rbrace$. Let $\alpha =\dim _H(E(\vec\{p\}))=\dim _P(E(\vec\{p\})) =\{\sum _\{j=0\}^\{m-1\} p_j\log p_j\over \sum _\{j=0\}^\{m- 1\}p_i \log r_j\}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\vec\{p\}=(r_0^s,r_1^s,\cdots ,r_\{m-1\}^s)$, then$\{\mathcal \{H\}\}^s(E(\vec\{p\}))=\{\mathcal \{H\}\}^s(E),\;\{\mathcal \{P\}\}^s(E(\vec\{p\}))=\{\mathcal \{P\}\}^s(E),$moreover both of $\{\mathcal \{H\}\}^s(E)$ and $\{\mathcal \{P\}\}^s(E)$ are finite positive;(ii) If $\vec\{p\}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_\{m-1\}^s)$, then the gauge functions can be partitioned as follows\begin\{equation*\} \{\mathcal \{H\}\}^g(E(\vec\{p\}))=+\infty \iff \mathop \{\overline\{\rm lim\}\}\_\{t\rightarrow 0\}\{\log g(t)\over \log t\}\le \alpha ;\ \{\mathcal \{H\}\}^g(E(\vec\{p\}))=0 \Longleftrightarrow \mathop \{\overline\{\rm lim\}\}\_\{t\rightarrow 0\}\{\log g(t)\over \log t\}&gt;\alpha , \end\{equation*\}\begin\{equation*\} \{\mathcal \{P\}\}^g(E(\vec\{p\}))=+\infty \Longleftrightarrow \mathop \{\underline\{\rm lim\}\}\_\{t\rightarrow 0\}\{\log g(t)\over \log t\}\le \alpha ;\ \{\mathcal \{P\}\}^g(E(\vec\{p\}))=0 \Longleftrightarrow \mathop \{\underline\{\rm lim\}\}\_\{t\rightarrow 0\}\{\log g(t)\over \log t\}&gt;\alpha . \end\{equation*\}},
affiliation = {Wuhan University, Department of Mathematics, Wuhan 430072 (Rép. Pop. Chine); Tsinghua University, Department of mathematics, Beijing 10084 (Rép. Pop. Chine); Wuhan University, Department of Mathematics, Wuhan 430072 (Rép. Pop. Chine)},
author = {Ma, Ji-Hua, Wen, Zhi-Ying, Wu, Jun},
journal = {Annales de l’institut Fourier},
keywords = {perturbation measures; gauge functions; Besicovitch set; Besicovitch sets; normal numbers; Hausdorff measures},
language = {eng},
number = {4},
pages = {1061-1074},
publisher = {Association des Annales de l'Institut Fourier},
title = {Besicovitch subsets of self-similar sets},
url = {http://eudml.org/doc/116003},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Ma, Ji-Hua
AU - Wen, Zhi-Ying
AU - Wu, Jun
TI - Besicovitch subsets of self-similar sets
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 4
SP - 1061
EP - 1074
AB - Let $E$ be a self-similar set with similarities ratio $r_j (0\le j\le m-1)$ and Hausdorff dimension $s$, let $\vec{p}(p_0,p_1)\ldots p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$E(\vec{p})=\Big \lbrace x\in E\colon \lim _{n\rightarrow \infty }{1\over n} \sum _{k=1}^{n}\chi _{j}(x_k)=p_j,\quad 0\le j\le m- 1\Big \rbrace ,$where $\chi _j$ is the indicator function of the set $\lbrace j\rbrace$. Let $\alpha =\dim _H(E(\vec{p}))=\dim _P(E(\vec{p})) ={\sum _{j=0}^{m-1} p_j\log p_j\over \sum _{j=0}^{m- 1}p_i \log r_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\vec{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then${\mathcal {H}}^s(E(\vec{p}))={\mathcal {H}}^s(E),\;{\mathcal {P}}^s(E(\vec{p}))={\mathcal {P}}^s(E),$moreover both of ${\mathcal {H}}^s(E)$ and ${\mathcal {P}}^s(E)$ are finite positive;(ii) If $\vec{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows\begin{equation*} {\mathcal {H}}^g(E(\vec{p}))=+\infty \iff \mathop {\overline{\rm lim}}_{t\rightarrow 0}{\log g(t)\over \log t}\le \alpha ;\ {\mathcal {H}}^g(E(\vec{p}))=0 \Longleftrightarrow \mathop {\overline{\rm lim}}_{t\rightarrow 0}{\log g(t)\over \log t}&gt;\alpha , \end{equation*}\begin{equation*} {\mathcal {P}}^g(E(\vec{p}))=+\infty \Longleftrightarrow \mathop {\underline{\rm lim}}_{t\rightarrow 0}{\log g(t)\over \log t}\le \alpha ;\ {\mathcal {P}}^g(E(\vec{p}))=0 \Longleftrightarrow \mathop {\underline{\rm lim}}_{t\rightarrow 0}{\log g(t)\over \log t}&gt;\alpha . \end{equation*}
LA - eng
KW - perturbation measures; gauge functions; Besicovitch set; Besicovitch sets; normal numbers; Hausdorff measures
UR - http://eudml.org/doc/116003
ER -

## References

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1. A.S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann 110 (1934), 321-330 Zbl0009.39503MR1512941
2. H.G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser 20 (1949), 31-36 Zbl0031.20801MR31026
3. K.J. Falconer, Techniques in Fractal Geometry, (1997), John Wiley and sons inc. Zbl0869.28003MR1449135
4. R. Kaufman, A further example on scales of Hausdorff functions, J. London Math. Soc 8 (1974), 585-586 Zbl0302.28015MR357721
5. M. Moran, J. Rey, Singularity of self-similar measures with respect to Hausdorff measures, Trans. of Amer. Math. Soc. 350 (1998), 2297-2310 Zbl0899.28002MR1475691
6. Y. Peres, The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure, Math. Proc. Camb. Phil. Soc 116 (1994), 513-526 Zbl0811.28005MR1291757
7. A.N. Shiryayev, Probability, (1984), Springer-Verlag, New York Zbl0536.60001MR737192
8. J. Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philo. Soc 100 (1986), 383-408 Zbl0622.60021MR857718

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