Abstract
In this paper we study properties of the fundamental domain F of number systems in the n-dimensional real vector space. In particular we investigate the fractal structure of its boundary F. In a first step we give upper and lower bounds for its box counting dimension. Under certain circumstances these bounds are identical and we get an exact value for the box counting dimension. Under additional assumptions we prove that the Hausdorf dimension of F is equal to its box counting dimension. Moreover, we show that the Hausdorf measure is positive and fnite. This is done by applying the theory of graphdirected self similar sets due to Falconer and Bandt. Finally, we discuss the connection to canonical number systems in number felds, and give some numerical examples.
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Müller, W., Thuswaldner, J.M. & Tichy, R.F. Fractal properties of number systems. Periodica Mathematica Hungarica 42, 51–68 (2001). https://doi.org/10.1023/A:1015292422840
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DOI: https://doi.org/10.1023/A:1015292422840