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Fractal properties of number systems. (English) Zbl 0980.11007

Let \(B\) be an \(n\times n\) matrix with integral coefficients and assume that all its eigenvalues lie outside the unit disc. Let \(0\in {\mathcal N} \subset \mathbb Z^n\) be a complete residue system \(\bmod B\). If \[ \mathbb Z^n=\Bigl\{\sum_jB^ja_j:\;a_j\in{\mathcal N}\Bigr\} \] then the pair \((B,{\mathcal N})\) is called a number system in \(\mathbb Z^n\). The fundamental domain \(F\) of such a system is defined by \[ F=\Bigl\{\sum_jB^{-j}a_j:\;a_j\in {\mathcal N}\Bigr\}. \] The authors study the boundary \(\delta F\) of \(F\) and prove that its Hausdorff dimension is always finite and positive. Actually their results apply to a more general class of pairs \((B,{\mathcal N})\), called “just touching covering systems”, characterized by the property that the intersection of two shifts of \(F\) by points of \(\mathbb Z^n\) is of zero measure. Applications to canonical number systems in imaginary quadratic fields, introduced by I. Kátai and J. Szabó [Acta Sci. Math. 37, 255-260 (1975; Zbl 0309.12001)], are also given.

MSC:

11A63 Radix representation; digital problems
28A78 Hausdorff and packing measures
28A80 Fractals
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