A Survey on Topological Properties of Tiles Related to Number Systems

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.     

On The Sum of Digits Function for Number Systems with Negative Bases

Let q 2 be an integer. Then –q gives rise to a number system in , i.e., each number n has a unique representation of the form n = c ₀ + c ₁ (–q) + ... + c _h (–q) ^h , with c _i {0,..., q – 1}(0 i h). The aim of this paper is to investigate the sum of digits function _–q (n) of these number systems. In particular, we derive an asymptotic expansion for

Topology of crystallographic tiles

We study self-affine tiles which tile the n-dimensional real vector space with respect to a crystallographic group. First we define classes of graphs that allow to determine the neighbors of a given tile algorithmically. In the case of plane tiles these graphs are used to derive a criterion for such tiles to be homeomorphic to a disk. As particular application, we will solve a problem of Gelbrich, who conjectured that certain examples of tiles which tile with respect to the ornament group p2 are homeomorphic to a disk.      

On a variant of the Kakeya problem in {\mathbb{R}}

We prove that the sharp lower bounds of the Minkowski and Hausdorff dimensions of circular Kakeya sets in ${\mathbb{R}}$ are 1/2 and 0 respectively.     

Fractals and Number Systems in Real Quadratic Number Fields

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

Patterns in Rational Base Number Systems

Number systems with a rational number a/b>1 as base have gained interest in recent years. In particular, relations to Mahler’s $\frac{3}{2}$ -problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base a/b and use representations w.r.t. this base to construct normal numbers in base a in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the adèle ring $\mathbb{A}_{\mathbb{Q}}$ and Fourier analysis in $\mathbb{A}_{\mathbb{Q}}$ . With help of these tools we are able to reformulate our results as estimation problems for character sums.     

On linear combinations of units with bounded coefficients and double-base digit expansions

Let $\mathfrak o $ be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in $\mathfrak o $ is the sum of pairwise distinct units, if the unit equation $u+v=2$ has a non-trivial solution $u,v\in \mathfrak o ^*$ . We generalize this result and give applications to signed double-base digit expansions.     

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).