## 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

### Keywords:

Hausdorff dimension; coordinate $$d$$-dimension print
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