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• W2059098568 abstract "An integral self-affine tile is the solution of a set equation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A script upper T equals union Underscript d element-of script upper D Endscripts left-parenthesis script upper T plus d right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>T</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>⋃<!-- ⋃ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>D</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>T</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {A} mathcal {T} = bigcup _{d in mathcal {D}} (mathcal {T} + d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n times n> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>n times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> integer matrix and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper D> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite subset of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Z Superscript n> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {Z}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A element-of double-struck upper Q Superscript n times n> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {A} in mathbb {Q}^{n times n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We define rational self-affine tiles as compact subsets of the open subring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n Baseline times product Underscript German p Endscripts upper K Subscript German p> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:munder> <mml:mo>∏<!-- ∏ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}^ntimes prod _mathfrak {p} K_mathfrak {p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the adèle ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper A Subscript upper K> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>A</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {A}_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where the factors of the (finite) product are certain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German p> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic completions of a number field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is defined in terms of the characteristic polynomial of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript n Baseline times product Underscript German p Endscripts StartSet 0 EndSet asymptotically-equals double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:munder> <mml:mo>∏<!-- ∏ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> </mml:munder> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>≃<!-- ≃ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}^n times prod _mathfrak {p} {0} simeq mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper D> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>D</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems." @default.
• W2059098568 created "2016-06-24" @default.
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• W2059098568 date "2015-03-13" @default.
• W2059098568 modified "2023-10-12" @default.
• W2059098568 title "Rational self-affine tiles" @default.
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