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Zbl 0715.11038
Ben Nasr, Fathi
Dimension d'ensembles plans définis par des propriétés des développements des coordonnées. (Dimension of plane sets, defined by expansion properties of the coordinates).
(French)
[J] Bull. Soc. Math. Fr. 118, No.1, 55-65 (1990). ISSN 0037-9484

Let $2\le r\sb 1<r\sb 2$ be any fixed integers. For any $x=(x\sp{(1)},x\sp{(2)})\in [0,1[\times [0,1[$ consider the expansions $x\sp{(1)}=\sum\sp{\infty}\sb{n=1}x\sb n/r\sp n\sb 1$, $x\sp{(2)}=\sum\sp{\infty}\sb{n=1}y\sb n/r\sp n\sb 2$, where $x\sb n\in N\sb{r\sb 1}=\{0,1,...,r\sb 1-1\}$, $y\sb n\in N\sb{r\sb 2}=\{0,1,...,r\sb 2-1\}$, $n\ge 1$, and put, for any $\alpha,\gamma \in N\sb{r\sb 1}$, $\beta,\delta \in N\sb{r\sb 2}$, $$N\sb{\alpha \beta,\gamma \delta}(x,n)=\#\{1\le i\le n: (x\sb i,y\sb i)=(\alpha,\beta),\quad (x\sb{i+1},y\sb{i+1})=(\gamma,\delta)\}.$$ For a given $r\sb 1r\sb 2\times r\sb 1r\sb 2$ matrix $M=(m\sb{\alpha \beta,\gamma \delta})$ satisfying the conditions $$(I)\quad m\sb{\alpha \beta,\gamma \delta}>0,\quad \alpha,\gamma \in N\sb{r\sb 1},\quad \beta,\delta \in N\sb{r\sb 2},\quad (II)\quad \sum\sp{r\sb 1-1}\sb{\alpha =0}\sum\sp{r\sb 1-1}\sb{\gamma =0}\sum\sp{r\sb 2-1}\sb{\beta =0}\sum\sp{r\sb 2-1}\sb{\delta =0}m\sb{\alpha \beta,\gamma \delta}=1,$$ $$(III)\quad \sum\sp{r\sb 1-1}\sb{\gamma =0}\sum\sp{r\sb 2-1}\sb{\delta =0}m\sb{\alpha \beta,\gamma \delta}=\sum\sp{r\sb 1-1}\sb{\gamma =0}\sum\sp{r\sb 2-1}\sb{\delta =0}m\sb{\gamma \delta,\alpha \beta},\quad \alpha \in N\sb{r\sb 1},\quad \beta \in N\sb{r\sb 2}$$ consider the set $$A=\{x\in [0,1[\times [0,1[ : \lim\sb{n\to \infty}(1/n)\quad N\sb{\alpha \beta,\gamma \delta}(x,n)=m\sb{\alpha \beta,\gamma \delta}\text{ for all } (\alpha,\beta)\text{ and } (\gamma,\delta)\}.$$ The author obtains an explicit expression for the Hausdorff dimension dim A of A provided M satisfies an additional condition. Assuming only (I)- (III), he obtains upper and lower bounds for dim A.
[K.Schürger]
MSC 2000:
*11K55 Metric theory of other number-theoretic algorithms and expansions
11K16 Normal numbers, etc.
28A78 Hausdorff measures
28A80 Fractals

Keywords: radix expansions of coordinates; digital problems; Hausdorff dimension

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