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Coordinate \(d\)-dimension prints. (English) Zbl 0819.28007

For \(E\subset \mathbb{R}^ 2\) and \(s,t>0\) set \[ CD^{(s,t)} (E)= \lim_{\delta\downarrow 0} \;\inf_{\delta>0} \{N_ E (a,b) s^ s b^ t:\;0<b\leq a\leq \delta\}, \] where \(N_ E (a,b)\) is the number of rectangles \([mb, (m+1) b]\times [na, (n+1) a]\), \(n,m\in \mathbb{Z}\), that intersect \(E\). The corresponding measure is defined by \[ cd^{s,t} (E)= \inf \Biggl\{ \sum_{n=1}^ \infty CD^{(s,t)} (E_ n):\;E= \bigcup_{n=1}^ \infty E_ n \Biggr\}. \] The authors introduce the set \[ \text{cd-Print} (E)= \{(s,t):\;cd^{(s,t)} (E)>0\}, \] called the coordinate \(d\)-dimension print of \(E\subset \mathbb{R}^ 2\). This concept allows to distinguish between sets with the same Hausdorff dimension. The authors study properties of the coordinate \(d\)-dimension print and demonstrate how to calculate it for different sets of interest.

MSC:

28A80 Fractals
28A78 Hausdorff and packing measures
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