Introduction Let K be a numberfield defined by a monic irreducible polynomial f ( x ) ∈ℤ[ x ] and ℤ K its ring of integers. It is well know that the ring ℤ K is a free ℤ-module of rank n = [ K :ℚ]. Thus the abelian group ℤ K /ℤ[α] is finite
In this paper we compute the box counting dimension of sets, that are related to number systems in real quadratic number fields.
The sets under discussion are so-called graph-directed self affine sets. Contrary to the case of self similar sets, for self
affine sets there does not exist a general theory for the determination of the box counting dimension. Thus we are forced
to construct the covers, needed for its calculation, directly.
In this paper, we present several methods for the construction of elliptic curves with large torsion group and positive rank
over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization