Integral self-affine tiles

The new setting

The irreducible caseSome news on rational self-affine tiles
J. M. Thuswaldner
joint work with J. Jankauskas and W. Steiner
Department of Mathematics and Statistics
University of Leoben
Austria

Paris, May 20182019

Integral self-affine tiles

The new setting

The irreducible case

Integral self-affine tiles
Definition
Let A be an expanding d × d integer matrix and D ⊂ Zd . The
non-empty compact set F = F(A, D) defined by[(F + d)
AF =d∈Dis called integral self-affine tile, if its d-dimensionalLebesgue-measure is positive.
Integral self-affine tiles were studied extensively e.g. by
R. Kenyon, K. Gröchenig, A. Haas, J. Lagarias, Y. Wang, andothers
W.l.o.g. we always assume that 0 ∈ D.

2019Integral self-affine tiles

The new setting

The irreducible case

Knuth’s twin dragonExample


0 −2
A=,
1 −2   01D=,
00This self-affine tile was studied by D. Knuth in connection withnumber systems defined in the ring Z[i].

2019Integral self-affine tilesThe new setting

The irreducible caseTilings
Lagarias and Wang proved that F + Zd forms a multiple tiling ofRd . In particular λd (F) ∈ N.
Theorem (Lagarias and Wang 1997)
Let A be an integer matrix with irreducible characteristic
polynomial and let D be a primitive standard digit set. ThenF(A, D) tiles Rd with respect to the lattice Zd .
Proof: Based on a Fourier analytic tiling criterion of Gröchenig
and Haas and on a result of Cerveau, Conze and Raugi ontransfer operators.
In the case of reducible matrices the theorem is no longer truein general. Lagarias and Wang gave a characterization in thiscase.

2019Integral self-affine tiles

The new setting

The irreducible case

Basic definitions

For A ∈ Qd×d let
Zd [A] =[(Zd + AZd + · · · + Ak −1 Zd )k ≥0

We define a standard digit set D as a complete set of coset
representatives ofZd [A]/AZd [A](without loss of generality, it can be assumed that 0 ∈ D).

The pair (A, D) is the data we need to define our mainobjects.

2019Integral self-affine tilesThe new setting

The irreducible caseA−1 -adic stuff
Let E ⊂ Zd be the set of coset representatives ofZd [A−1 ]/A−1 Zd [A−1 ]and setEk := Zd [A−1 ]/A−k Zd [A−1 ]A−1 -adic integers ZA−1 as projective limit: ZA−1 = lim Ek .
←−A−1 -adicSrationals are then is defined by
QA−1 = j≥0 Aj ZA−1 ;P
A−1 -adic norm: for x = kj=−∞ Aj cj with ck 6= 0,||x||A−1 = |E|k ,Haar measure µA−1 in QA−1 , such that
µA−1 (x + Ak ZA−1 ) = |E|kfor every x ∈ QA−1 .

2019

Integral self-affine tiles

The new setting

The irreducible case

Representation space: where the tiles live
Representation space KA−1 = Rd × QA−1 .
The diagonal embeddingΦA−1 : Zd [A, A−1 ] → KA−1
maps x ∈ Zd [A, A−1 ] into ΦA−1 (x) = (x, x).
The multiplication by A in KA−1 : A · (x, y) = (Ax, Ay).
Product norm:||x|| := ||x||K = max{||x||∞ , ||x||A−1 }Product measure: µK = µ∞ ⊗ µA−1By the definition of the measure µK , for every measurable
set M ⊂ KA−1 , one hasµK (A · M) = |det A| · |E| · µK (M).

2019

Integral self-affine tiles

The new setting

The irreducible case

Rational self-affine tiles

The main object of our investigations is a compact set
F = F(A, D) ⊂ KA−1 that satisfies the equation[F + (d, d) .
AF =d∈D

The set F is called rational self-affine tile if µK (F) > 0.
Lemma
The set ΦA−1 (Zd [A]) is a uniformly discrete and relatively dense
subgroup of KA−1 .

2019

Integral self-affine tiles

The new setting

The irreducible case

An example ...

A = 23 and D = {0, 1, 2}.

Representation space K 2 = R × Q2 .
3

F = F( 23 , {0, 1, 2}) is a compact subset of K 2 ,3

F = int(F) and µ 3 (∂F) = 0.
2

{F + Φ 2 (x) : x ∈ Z[ 32 ]}, forms a tiling of K 2 .
33

2019

Integral self-affine tiles

The new setting

The irreducible case2019... with a picture

−3/2

1/2

−5/2

5/2

−1/2

3/2−1G(−2)

7/2

1−20

G(0)

9/2

3524G(2)G(4)

PHere, an element ∞bj α−j of Q2 , with bj ∈ {0, 1}, isP∞ j=k −jrepresented by j=k bj 2 .

Integral self-affine tiles

The new setting

The irreducible case

Properties of rational self-affine tiles

Theorem (Jankauskas, Steiner, and T. (2018+))
Let A ∈ Qd×d be an expanding rational matrix and let
D ⊂ Zd [A] be a standard digit set. Then the following properties
hold for the rational self-affine tile F = F(A, D).

1) F is a tile and F o 6= ∅.

2) F is the closure of its interior, i.e. F = F o .

3) The boundary ∂F of F has measure µK (∂F) = 0.

4) The collection of sets {F + ΦA−1 (z) : z ∈ Zd [A]} is amulti-tiling of KA−1 .

2019

Integral self-affine tiles

The new setting

The irreducible case

“Adelic” definition of rational self affine tiles
Definition (Steiner and T. (2015))
Let α be a root of the characteristic polynomial and let (r , s) be
the signature of Q(α). Choose D ⊂ Z[α].

Let (α) = ba where (a, b) = O and define the representation
spaceYY
Kα =Kp = Rr × Cs ×Kp .
p|∞ or p|bp|b

The rational self-affine tile F = Fα = F(A, D) is defined by(if its Haar measure in Kα is positive)[αFα =(Fα + Φα (d)).
d∈DΦα is the diagonal embedding of Q(α) in Kα .

2019

Integral self-affine tiles

The new setting

The irreducible case

Remarks on the definition
Haar Measure
The Haar measure µ on Kα is the product of the Haar
measures µp on Kp . For finite p we set µp (a + pm ) = N(p)−m , for
infinite primes µp is the Lebesgue measure on Kp (Kp = R or C).
Embedding of Q(α)Q(α) is naturally embedded in Kα diagonally via
Φ(ξ) = (α, . . . , α). Moreover, Q(α) acts multiplicatively on thering Kα , in particular ξ · z = Φα (ξ)z.
Tiling of Rd
One could also define Fα as a subset of Rd as in the integral
case. However, this would result in sets that have no nice tilingproperties.

2019Integral self-affine tiles

The new setting

The irreducible case

Relation between the two definitions
Choose an expanding matrix A ∈ Qd×d with irreduciblecharacteristic polynomial.
Let α be a root of the characteristic polynomial of A.
Choose a basis of Q(α) (viewed as a vector space over Q)
such that the multiplication by α is done by A in this vector
space.
Chose a subset D ⊂ Z[α] as set of digits.

An equivalent definition for a rational self-affine tile F ⊂ Kα
is given by the set equation[AF =(F + Φα (d)).
d∈D2019

Integral self-affine tiles

The new setting

The irreducible caseSpecial digit sets
Again we need special properties of the digit set.
Definition
Let α be expanding and D ⊂ Z[α] be given.
The digit set D is called primitive ifhD, αD, α2 D, . . .iZ = Z[α].
The digit set D is called standard digit set if D is acomplete set of residues of Z[α]/αZ[α].
Set Zhα, Di = hD, αD, α2 D, . . .iZ

2019Integral self-affine tiles

The new setting

The irreducible case

Tiling theorems
Much harder to prove is the following tiling theorem.
Theorem (Steiner and T. (2015))
Let α be an expanding algebraic number and let D be astandard digit set for α. Then {F + Φα (x) : x ∈ Zhα, Di} forms
a tiling of Kα .
For primitive digit sets, we get the following immediate corollary.
Corollary (Steiner and T. (2015))
Let α be an expanding algebraic number and let D be aprimitive, standard digit set for α. Then {F + Φα (x) : x ∈ Z[α]}forms a tiling of Kα .
Note that for instance {0, 1} ∈ D implies primitivity of the digitset.

2019Integral self-affine tiles

The new setting

The irreducible caseThank you for your attention

2019Integral self-affine tiles

The new setting

The irreducible case

Numeration 2019

Numeration 2019
Erwin Schrödinger Institute
Vienna, July 8 - 12, 2019

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2019