Number Systems and Fractal Geometry

Indlekofer, Karl-Heinz; Racskó, P.; Катаи, И.

doi:10.1007/978-94-011-2817-9_21

Cited by 22 publications

(7 citation statements)

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“…Our construction will also give a clear characterisation of simultaneous number systems, as defined in [5]. Among others, we obtain an easy proof of the following, where we use a recent theorem on products of linear polynomials independently due to Kane [6] and Pethő [11].…”

Section: Introductionmentioning

confidence: 89%

Number systems and the Chinese Remainder Theorem

van de Woestijne

2011

Preprint

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A well-known generalisation of positional numeration systems is the case where the base is the residue class of x modulo a given polynomial f (x) with coefficients in (for example) the integers, and where we try to construct finite expansions for all residue classes modulo f (x), using a suitably chosen digit set. We give precise conditions under which direct or fibred products of two such polynomial number systems are again of the same form. The main tool is a general form of the Chinese Remainder Theorem. We give applications to simultaneous number systems in the integers.

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Section: Introductionmentioning

confidence: 89%

Number systems and the Chinese Remainder Theorem

van de Woestijne

2011

Preprint

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“…Firstly, a, b, c are associated with 1, 2, 3, andȧ,ḃ,ċ with 4, 5, 6, since the underlying elements of R are exactly P, Q, N, −P, −Q, −N . Secondly, 7,8,9,10,11,12 are associated with c, b, a,ċ,ḃ,ȧ respectively, as the states 7, . .…”

Section: The Relation To the Recurrent Set Methodsmentioning

confidence: 99%

“…Thus the boundary of T is the attractor of this graph iterated function system (see [8,20]). G(R) is usually smaller than the boundary GIFS encountered in the literature [12,14,23,28]. Indeed, the set of states of the latter consists of all the neighbors of T , that is, the tiles T + s with T ∩ (T + s) = ∅.…”

Section: G(r) and The Boundary Parametrizationmentioning

confidence: 99%

Boundary parametrization of planar self-affine tiles with collinear digit set

Akiyama

Loridant

2010

Sci. China Math.

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We consider a class of planar self-affine tiles T = M −1 a∈D (T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:We give a parametrization S 1 → ∂T of the boundary of T with the following standard properties. It is Hölder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.

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“…Then k ∈ S if and only if x m = k for some m (2ω k + 1) n , where U , B and ω k were given in the statements just before Lemma 2.1 After we obtained Theorem 2.1, we became aware that Indlekofer, Kátai and Racskó had a theorem [12,Theorem 1] in the special case of complex number systems, corresponding to a special class of 2 × 2 integral expansive matrices and certain digit sets, early in 1992, which is analogous to our Theorem 2.1. We also found out that Kovacs had a related result for general integral expansive matrices from which the inclusion S ⊂ −K ∩ Z n can be deduced [15,Theorem 2].…”

Section: Introductionmentioning

confidence: 99%

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

Gabardo

2006

J. London Math. Soc.

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In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice Γ ⊆ Z n as a translation tiling set of R n . In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile K and to determine all of the lattice tilings of R n by K. Moreover, we also propose and determine algorithmically another type of translation tiling of R n by K, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set K ∩ (K + j), j ∈ Z n , is strictly positive. Proposition 1.1 [10, 17, 19]. The Lebesgue measure of any integral self-affine tile K(B, D) is a positive integer.However, an efficient algorithm to evaluate this measure is not yet available, to our knowledge. We say that a measurable set E ⊂ R n tiles R n using the discrete translation tiling set T if t∈T χ E+t = 1 almost everywhere on R n , where χ A denotes the characteristic function of A. In this case, we also say that E + T is a tiling of R n . One also knows that certain sets of integral translates of K tile R n (see [8,10,16]). In particular, Lagarias and Wang [18] as well as Conze, Hervé and Raugi [4] independently proved that every self-affine tile admits a lattice tiling.Proposition 1.2 [4, 18]. Every integral self-affine tile K(B, D) admits some lattice Γ ⊆ Z n as a tiling set for R n . Natural tiling and Lebesgue measure of K(B, D)Our first goal is to give a characterization for the integral points in K, which will be needed later on. Before doing so, we need to introduce some terminology.Definition 2.1. Given an integral expansive matrix B and associated standard digit set D, with 0 ∈ D, we have that, for any k ∈ Z n , and every m 11) where d 0 , . . . , d m−1 ∈ D, x m ∈ Z n . So for each k ∈ Z n , we can associate a unique sequence (d 0 , d 1 , d 2 , . . . ), and formally write k ∼ (d 0 , d 1 , d 2 , . . . ).It is easy to prove that, for each given k ∈ Z n , the sequence {x m : m 1} ⊂ Z n in (2.1) is bounded and can, therefore, take only finite distinct values.

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Number Systems and Fractal Geometry

Cited by 22 publications

References 0 publications

Number systems and the Chinese Remainder Theorem

Number systems and the Chinese Remainder Theorem

Boundary parametrization of planar self-affine tiles with collinear digit set

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

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