Riesz bases, Meyers quasicrystals, and bounded remainder sets - PowerPoint PPT Presentation

Riesz bases, Meyer’s quasicrystals, and bounded remainder sets Sigrid Grepstad June 7, 2018 Joint work with Nir Lev

Riesz bases of exponentials S ⊂ R d is bounded and measurable. Λ ⊂ R d is discrete. The exponential system e λ ( x ) = e 2 πi � λ,x � , E (Λ) = { e λ } λ ∈ Λ , is a Riesz basis in the space L 2 ( S ) if the mapping f → {� f, e λ �} λ ∈ Λ is bounded and invertible from L 2 ( S ) onto ℓ 2 (Λ) .

Known results Kozma, Nitzan (2012): Finite unions of intervals Kozma, Nitzan (2015): Finite unions of rectangles in R d G., Lev and Kolountzakis (2012/2013): Multi-tiling sets in R d Lyubarskii, Rashkovskii (2000): Convex, centrally symmetric polygons in R 2 Questions What about the ball in dimensions two and higher? Does every set in R d admit a Riesz basis of exponentials?

Density Lower and upper uniform densities: #(Λ ∩ ( x + B R )) D − (Λ) = lim inf inf | B R | R →∞ x ∈ R d #(Λ ∩ ( x + B R )) D + (Λ) = lim sup sup | B R | R →∞ x ∈ R d If E (Λ) is a Riesz basis in L 2 ( S ) , then D − (Λ) = D + (Λ) = mes S .

Cut-and-project sets R n Γ R m

Cut-and-project sets R n Γ W R m

Cut-and-project sets R n Γ W R m

Cut-and-project sets R n Γ W R m

Cut-and-project sets R n Γ W R m We define the Meyer cut-and-project set Λ(Γ , W ) = { p 1 ( γ ) : γ ∈ Γ , p 2 ( γ ) ∈ W } , with density D (Λ) = mes W/ det Γ .

Simple quasicrystals R Γ I R d We define the simple quasicrystal Λ(Γ , I ) = { p 1 ( γ ) : γ ∈ Γ , p 2 ( γ ) ∈ I } , with density D (Λ) = | I | / det Γ .

Sampling on quasicrystals Matei and Meyer (2008): Simple quasicrystals are universal sampling sets. Kozma, Lev (2011): Riesz bases of exponentials from quasicrystals in dimension one.

Theorem 1 Let Λ = Λ(Γ , I ) , and suppose that | I | / ∈ p 2 (Γ) . Then there exists no Riemann measurable set S such that E (Λ) is a Riesz basis in L 2 ( S )

Equidecomposability 3 2 1 1 2 3 S S’ The sets S and S ′ are equidecomposable (or scissors congruent).

Theorem 2 Let Λ = Λ(Γ , I ) , and suppose that | I | ∈ p 2 (Γ) . Then E (Λ) is a Riesz basis in L 2 ( S ) for every Riemann measurable set S , mes S = D (Λ) , satisfying the following condition: S is equidecomposable to a parallelepiped with vertices in p 1 (Γ ∗ ) , using translations by vectors in p 1 (Γ ∗ ) . Γ ∗ = � γ ∗ ∈ R d × R : � γ, γ ∗ � ∈ Z for all γ ∈ Γ �

Example 1 Let α be an irrational number, and define Λ = { λ ( n ) } by λ ( n ) = n + { nα } , n ∈ Z . Then E (Λ) is a Riesz basis in L 2 ( S ) for every S ⊂ R , mes S = 1 , which is a finite union of disjoint intervals with lengths in Z α + Z .

Example 1 Let α be an irrational number, and define Λ = { λ ( n ) } by λ ( n ) = n + { nα } , n ∈ Z . Then E (Λ) is a Riesz basis in L 2 ( S ) for every S ⊂ R , mes S = 1 , which is a finite union of disjoint intervals with lengths in Z α + Z . Notice that { λ ( n ) } n ∈ Z = Λ(Γ , I ) , where I = [0 , 1) and Γ = { ((1 + α ) n − m, nα − m ) : m, n ∈ Z } , Γ ∗ = { ( nα + m, − n (1 + α ) − m ) : m, n ∈ Z }

Example 2 Let Λ = { λ ( n, m ) } be defined by √ √ √ √ ( n, m ) ∈ Z 2 . λ ( n, m ) = ( n, m ) + { n 2 + m 3 } ( 2 , 3) , E (Λ) is a Riesz basis in L 2 ( S ) for every set S ⊂ R 2 which is equidecomposable to the unit cube [0 , 1) 2 using only translations by √ √ 3) + Z 2 . vectors in Z ( 2 ,

Corollary 1 Λ = Λ(Γ , I ) , | I | ∈ p 2 (Γ) K ⊂ R d compact, U ⊂ R d open K ⊂ U and mes K < D (Λ) < mes U There exists a set S ⊂ R d satisfying: i) K ⊂ S ⊂ U and mes S = D (Λ) . ii) S is equidecomposable to a parallelepiped with vertices in p 1 (Γ ∗ ) using translations by vectors in p 1 (Γ ∗ ) .

Duality Λ(Γ , I ) = { p 1 ( γ ) : γ ∈ Γ , p 2 ( γ ) ∈ I } ⊂ R d Λ ∗ (Γ , S ) = { p 2 ( γ ∗ ) : γ ∗ ∈ Γ ∗ , p 1 ( γ ∗ ) ∈ S } ⊂ R Duality lemma Suppose that E (Λ ∗ (Γ , S )) is a Riesz basis in L 2 ( I ) . Then E (Λ(Γ , I )) is a Riesz basis in L 2 ( S ) .

Lattices of special form �� � � (Id + βα ⊤ ) m − βn, n − α ⊤ m : m ∈ Z d , n ∈ Z Γ = Γ ∗ = �� � � m + nα, (1 + β ⊤ α ) n + β ⊤ m : m ∈ Z d , n ∈ Z Theorem 2 Let Λ = Λ(Γ , I ) and suppose that | I | = m 1 α 1 + · · · m d α d + n for integers m 1 , . . . , m d and n . Then E (Λ) is a Riesz basis in L 2 ( S ) for every Riemann measurable set S , mes S = | I | , which is equidecomposable to a parallelepiped with vertices in Z d + α Z using translations by vectors in Z d + α Z .

By duality, we may choose to consider � � Λ ∗ (Γ , S ) = n + β ⊤ ( nα + m ) : nα + m ∈ S , where n ∈ Z and m ∈ Z d . Question: When is E (Λ ∗ ) a Riesz basis in L 2 ( I ) for an interval of length | I | = mes S ?

Avdonin’s theorem Avdonins theorem Let I ⊂ R be an interval and Λ = { λ j : j ∈ Z } be a sequence in R satisfying: (a) Λ is separated; (b) sup j | δ j | < ∞ , where δ j := λ j − j/ | I | ; (c) There is a constant c and positive integer N such that � � k + N � 1 � 1 � � � sup δ j − c < � � N 4 | I | k ∈ Z � � j = k +1 � � Then E (Λ) is a Riesz basis in L 2 ( I ) .

� � Λ ∗ (Γ , S ) = n + β ⊤ ( nα + m ) : n ∈ Z , m ∈ Z d , nα + m ∈ S � � � n + β ⊤ s : s = nα + m ∈ S = Λ n , Λ n = R S α R

Irrational rotation on the torus S ⊂ T d = R d / Z d α = ( α 1 , α 2 , . . . , α d ) The sequence { nα } is equidistributed. S n − 1 1 � χ S ( x + kα ) → mes S n k =0 ( n → ∞ ) α n − 1 � D n ( S, x ) = χ S ( x + kα ) − n mes S = o ( n ) k =0

Bounded remainder sets Definition A set S is a bounded remainder set (BRS) if there is a constant C = C ( S, α ) such that � n − 1 � � � � | D n ( S, x ) | = χ S ( x + kα ) − n mes S � ≤ C � � � � � k =0 for all n and a.e. x .

Claim: The quasicrystal Λ ∗ (Γ , S ) is at bounded distance from { j/ mes S } j ∈ Z if and only if S is a bounded remainder set.

Claim: The quasicrystal Λ ∗ (Γ , S ) is at bounded distance from { j/ mes S } j ∈ Z if and only if S is a bounded remainder set. Λ ∗ = � � n + β ⊤ ( nα + m ) : n ∈ Z , m ∈ Z d , nα + m ∈ S � � � n + β ⊤ s : s = nα + m ∈ S = Λ n , Λ n = n

Claim: The quasicrystal Λ ∗ (Γ , S ) is at bounded distance from { j/ mes S } j ∈ Z if and only if S is a bounded remainder set. 0 K Λ ∗ = � � n + β ⊤ ( nα + m ) : n ∈ Z , m ∈ Z d , nα + m ∈ S � � � n + β ⊤ s : s = nα + m ∈ S = Λ n , Λ n = n

Claim: The quasicrystal Λ ∗ (Γ , S ) is at bounded distance from { j/ mes S } j ∈ Z if and only if S is a bounded remainder set. 0 K Λ ∗ = � � n + β ⊤ ( nα + m ) : n ∈ Z , m ∈ Z d , nα + m ∈ S � � � n + β ⊤ s : s = nα + m ∈ S = Λ n , Λ n = n K − 1 K − 1 N = | Λ ∗ ∩ [0 , K ) | = � � | Λ k | + const = χ S ( kα ) + const k =0 k =0

Claim: The quasicrystal Λ ∗ (Γ , S ) is at bounded distance from { j/ mes S } j ∈ Z if and only if S is a bounded remainder set. 0 K Λ ∗ = � � n + β ⊤ ( nα + m ) : n ∈ Z , m ∈ Z d , nα + m ∈ S � � � n + β ⊤ s : s = nα + m ∈ S = Λ n , Λ n = n K − 1 K − 1 N = | Λ ∗ ∩ [0 , K ) | = � � | Λ k | + const = χ S ( kα ) + const k =0 k =0 = | Z / mes S ∩ [0 , K ) | + const = K mes S + const

Properties of bounded remainder sets Theorem (G., Lev 2015) Any parallelepiped in R d spanned by vectors v 1 , . . . , v d belonging to Z α + Z d is a bounded remainder set. (Duneau and Oguey (1990): Displacive transformations and quasicrystalline symmetries ) Theorem The measure of any bounded remainder set must be of the form n 0 + n 1 α 1 + · · · + n d α d where n 0 , . . . n d are integers.

Characterization of Riemann measurable BRS Theorem A Riemann measurable set S ⊂ R d is a BRS if and only if there is a parallelepiped P spanned by vectors belonging to Z α + Z d , such that S and P are equidecomposable using translations by vectors in Z α + Z d only.

Summary proof Theorem 2 Λ ∗ (Γ , S ) provides a Riesz basis E (Λ ∗ ) in L 2 ( I ) whenever S ⊂ R d is a bounded remainder set with mes S = | I | , i.e. if S is equidecomposable to a parallelepiped spanned by vectors in Z α + Z d using translations by vectors in Z α + Z d . ⇓ (Duality) Λ(Γ , I ) gives a Riesz basis E (Λ) in L 2 ( S ) for all such sets S . Note: The given equidecomposition condition on S implies that mes S = n 0 + n 1 α 1 + · · · + n d α d ∈ p 2 (Γ) .

Pavlov’s complete characterization One can deduce from Pavlov’s complete characterization of exponential Riesz bases in L 2 ( I ) that for Λ ∗ = Λ ∗ (Γ , S ) to provide a Riesz basis in L 2 ( I ) it is necessary that the sequence of discrepancies � n − 1 � � { d n } n ≥ 1 = χ S ( kα ) − n mes S k =0 n ≥ 1 is in BMO, i.e. satisfies m � � � � 1 � d k − d n +1 + · · · + d m � � � sup < ∞ . � � m − n m − n n<m � k = n +1