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Quasicrystals, bi-Lipschitz Equivalence and Bounded Movement.
Alexey Garber
Moscow State University

April 17, 2010

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Ё This work is a joint work with Dirk Frettloh from Bielefeld University.

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Separated net

Definition
A subset A of metric space M is called separated net if for some constants r and R following condition holds: for every two points x1 , x2 A distance dM (x1 , x2 ) r and for every point y M dM (y , A) R .

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Bi-Lipschitz equivalence

Definition
Two sets A1 M1 and A2 M2 in two possibly different metric spaces are called bi-Lipschitz equivalent if there exist a bijection f : A1 - A2 and a constant L 1 such that for any two points x , y A1 the following inequality holds: 1 · d A1 ( x , y ) d A2 ( f ( x ) , f ( y ) ) L · d A1 ( x , y ) . L Two bi-Lipschitz equivalent sets we will designate A1 A2 .
Lip

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Examples

Any two lattices are bi-Lipschitz equivalent.

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Examples

Any two lattices are bi-Lipschitz equivalent. Also any set is equivalent to its affine image.

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The problem of Gromov and Furstenberg

Problem
Find a practical criterion on a metric space M that would insure a bi-Lipschitz equivalence between every two separated nets in M .

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Some non-Euclidean results

It is evident that for Sd any two separated nets are equivalent if and only if they have the same cardinality.

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Some non-Euclidean results

It is evident that for Sd any two separated nets are equivalent if and only if they have the same cardinality. P. Papasoglu, 1995. Any two homogeneous trees of valence at least 3 are bi-Lipschitz equivalent.

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Some non-Euclidean results

It is evident that for Sd any two separated nets are equivalent if and only if they have the same cardinality. P. Papasoglu, 1995. Any two homogeneous trees of valence at least 3 are bi-Lipschitz equivalent. O. Bogopolskii, 1997. Any two separated nets in hyperbolic space H are equivalent.
d

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Euclidean results

It is evident that in R1 any two separated nets are equivalent.

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Euclidean results

It is evident that in R1 any two separated nets are equivalent. D. Burago, B. Kleiner and in the same time C. McMullen, 1998. There exist a separated net in Rd , d > 1 which is not equivalent to lattice.

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Bounded distance sets

Lemma
Assume that for two separated nets A and B there exist a bijection f : A - B and a constant > 0 such that for every point x A the inequality d(x , f (x )) holds. Then A B .
Lip

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Bounded distance sets

Lemma
Assume that for two separated nets A and B there exist a bijection f : A - B and a constant > 0 such that for every point x A the inequality d(x , f (x )) holds. Then A B .
Lip

In this case we will say that A and B are at bounded distance.

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Bounded distance sets

Lemma
Assume that for two separated nets A and B there exist a bijection f : A - B and a constant > 0 such that for every point x A the inequality d(x , f (x )) holds. Then A B .
Lip

In this case we will say that A and B are at bounded distance. For Hd O.Bogopolskii proved not only that any two Delone sets are bi-Lipschitz equivalent but they are also at bounded distance.

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Quasicrystals (model sets)

Definition
Let be a lattice in Rn в Rm , 1 : Rn в Rm Rn 2 : Rn в Rm Rm be projections, such that 1 | is dense in Rm . Let W Rm be a compact set t the closure of the interior of W equals W . This is summarized in the following diagram, which cut-and-project scheme.


and is injective, and 2 () he window such that is called

2 1 Rn - Rn в Rm - Rm V W

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Quasicrystals (model sets)

Definition
Then V := V (Rn , Rm , , W ) = {1 (x ) | x , 2 (x ) W } is called a canon The space Rn is If µ( (W )) = 0, If (W ) 2 () Also we can can ical model set. called physical space and Rm is called internal space then V is called regular model set. = , then V is called generic model set. consider a case with open bounded window W .

This definition can be generalized for any locally compact Abelian groups.

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Example of model set

Figure: Fionacci quasilattice V (R1 , R1 , Z2 , [0, 1)) with physical space b R1 : y = 5+1 x . 2

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Example of model set II

Figure: Penrose tiling.
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Quasicrystals and bounded movement

Theorem (M.Duneau, C.Oguey, 1990.)
If window W is a translation copy of a fundamental domain of 2 -projection of some m-sublattice of then the correspondent regular model set V (Rn , Rm , , W ) is at bounded distance from Zn .

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Lemmas of Duneau and Oguey.

Lemma
If 1 and 2 are two lattices in Rd of the same density then they are at the bounded distance.

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Lemmas of Duneau and Oguey.

Lemma
If 1 and 2 are two lattices in Rd of the same density then they are at the bounded distance.

Lemma
If 1 and 2 are two non-intersecting lattices in Rd of densities 1 and respectively then 1 2 is at bounded distance from any lattice with density 1 + 2 .
2

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Constructing more windows

This theorem can be used to obtain quasicrystals with more complicated windows which are at bounded distance from a lattice.

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Constructing more windows

This theorem can be used to obtain quasicrystals with more complicated windows which are at bounded distance from a lattice. Consider a quasicrystal V (R2 , R2 , Z4 , W ) where W is a regular octagon and basic vectors of Z4 projected on halves of diagonals of the octagon.

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Octagon as a union of fundamental domains

Figure: Octagon can not be a fundamental domain of any 2-lattice.

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Octagon as a union of fundamental domains

Figure: Hexagon is a fundamental domain of a sublattice.

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Octagon as a union of fundamental domains

Figure: Each of three parallelograms is a fundamental domain.

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Quasicrystals and bounded movement II

Problem
What types of windows will give us quasicrystals which are at bounded distance from lattices?

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Quasicrystals and bounded movement II

Problem
What types of windows will give us quasicrystals which are at bounded distance from lattices?

Theorem (H.Kesten, 1966)
The only quasicrystals V (R1 , R1 , , W ) that satisfy the condition of Duneau and Oguey (possibly with operations of union and subtraction of windows as a sets) are at bounded distance from some one-dimensional lattice.

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Quasicrystals and bi-Lipschitz equivalence
Consider a real space R3 and a model set V = V (R2 , R1 , Z3 , W ) where the physical space R2 is a plane : z = x + y .

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Quasicrystals and bi-Lipschitz equivalence
Consider a real space R3 and a model set V = V (R2 , R1 , Z3 , W ) where the physical space R2 is a plane : z = x + y .

Theorem (D. Burago, B. Kleiner, 2002.)
C If satisfies a condition - p > q d for some constants C > 0 and q d > 2 and every natural p , q then the set V is bi-Lipschitz equivalent to the lattice Z2 .

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Quasicrystals and bi-Lipschitz equivalence
Consider a real space R3 and a model set V = V (R2 , R1 , Z3 , W ) where the physical space R2 is a plane : z = x + y .

Theorem (D. Burago, B. Kleiner, 2002.)
C If satisfies a condition - p > q d for some constants C > 0 and q d > 2 and every natural p , q then the set V is bi-Lipschitz equivalent to the lattice Z2 .

Theorem (Y.Solomon, 2007.)
Delone set created from the centers of Penrose tiling is bi-Lipschitz equivalent to a lattice.

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Quasicrystals and bi-Lipschitz equivalence

Ё Theorem (D.Frettloh, A.Garber, 2009.)
Any two-dimensional regular generic model set is bi-Lipschitz equivalent to a lattice.

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