Local translations associated to spectral sets
Abstract.
In connection to the Fuglede conjecture, we study groups of local translations associated to spectral sets, i.e., measurable sets in or that have an orthogonal basis of exponential functions. We investigate the connections between the groups of local translations on and on and present some examples for low cardinality. We present some relations between the group of local translations and tilings.
Key words and phrases:
Spectrum, tile, Hadamard matrix, Fuglede conjecture, local translations2000 Mathematics Subject Classification:
42A16,05B45,15B34Contents
1. Introduction
In his study of commuting selfadjoint extensions of partial differential operators, Fuglede proposed the following conjecture [Fug74]:
Conjecture 1.1.
Denote by , . Let be a Lebesgue measurable subset of of finite positive measure. There exists a set such that is an orthogonal basis in if and only if tiles by translations.
Definition 1.2.
Let be a Lebesgue measurable subset of of finite, positive measure. We say that is a spectral set if there exists a set in such that is an orthogonal basis in . In this case, is called a spectrum for .
We say that tiles by translations if there exists a set in such that the sets , form a partition of , up to measure zero.
Terrence Tao [Tao04] has proved that spectraltile implication in the Fuglede conjecture is false in dimensions and later both directions were disproved in dimensions , see [KM06a]. At the moment the conjecture is still open in both directions for dimensions 1 and 2. In this paper, we will only focus on dimension .
Recent investigations have shown that the Fuglede can be reduced to analogous statements in , see [DL13].
Definition 1.3.
Let be a finite subset of , . We say that is spectral if there exists a set in such that is an orthogonal basis in , equivalently, the matrix
(1.1) 
is unitary. This matrix is called the Hadamard matrix associated to the pair .
We say that tiles by translations if there exists a set in such that the sets , forms a partition of .
The Fuglede conjecture for can be formulated as follows
Conjecture 1.4.
A finite subset of is spectral if and only if it tiles by translations.
As shown in [DL13] the tilespectral implications for and are equivalent:
Theorem 1.5.
Every bounded tile in is spectral if and only if every tile in is spectral.
It is not clear if the spectraltile implications for and are equivalent. It is known that, if every spectral set in is a tile, then every spectral set in is a tile in . It was shown in [DL13], that the reverse also holds under some extra assumptions.
Theorem 1.6.
Suppose every bounded spectral set in , of Lebesgue measure , has a rational spectrum, . Then the spectraltile implications for and for are equivalent, i.e., every bounded spectral set in is a tile if and only if every spectral set in is a tile.
All known examples of spectral sets of Lebesgue measure 1 have a rational spectrum. There is another, stronger variation of the spectraltile implication in which is equivalent to the spectraltile implication in .
Definition 1.7.
We say that a set has period if . The smallest positive with this property is called the minimal period.
Theorem 1.8.
[DL13] The following statements are equivalent

Every bounded spectral set in is a tile.

For every finite union of intervals with , if is a spectrum of with minimal period , then tiles with a tiling set .
We should note here that for a finite set in , the set is spectral in if and only if is spectral in . Also, if is a spectrum for then is a spectrum of , therefore has period 1 and the minimal period will have to be of the form . See [DL13] for details.
The spectral property of a set, in either or , can be characterized by the existence of a certain unitary group of local translations. We will describe this in the next section and present some properties of these groups. In Theorem 2.11 we give a characterization of spectral sets in in terms of the existence of groups of local translations or of a local translation matrix. In Proposition 2.14 we establish a formula that connects the local translation matrix for a spectral set in and the group of local translations for the spectral set in . Proposition 2.16 shows how one can look for tiling sets for the spectral set using the local translation matrix. Proposition 2.18 shows that the rationality of the spectrum is characterized by the periodicity of the group of local translations.
2. Local translations
In this section we introduce the unitary groups of local translations associated to spectral sets. These are oneparameter groups of unitary operators on , for subsets of , or on , for subsets of , which act as translations on or whenever such translations are possible. The existence of such groups was already noticed earlier by Fuglede [Fug74] and [Ped87]. They were further studied in [DJ12b]. The idea is the following: the existence of an orthonormal basis allows the construction of the Fourier transform from to . On one has the unitary group of modulation operators, i.e., multiplication by , or the diagonal matrix with entries , . Conjugating via the Fourier transform we obtain the unitary group of local translations. For further information on local translations and connections to selfadjoint extensions and scattering theory, see [JPT12a, JPT12d, JPT12b, JPT12c].
Definition 2.1.
Let be a bounded Borel subset of . A unitary group of local translations on is a strongly continuous one parameter unitary group on with the property that for any and any ,
(2.1) 
If is spectral with spectrum , we define the Fourier transform
(2.2) 
We define the unitary group of local translations associated to by
(2.3) 
Proposition 2.2.
With the notations in Definition 2.1
(2.4) 
Thus the functions are the eigenvectors for the operators corresponding to the eigenvalues with multiplicity one.
Proof.
Clearly for all . The rest follows from a simple computation. ∎
Theorem 2.3.
Let be a bounded Borel subset of . Assume that is spectral with spectrum . Let be the associated unitary group as in Definition 2.1. Then is a unitary group of local translations.
In the particular case when is a finite union of intervals the converse also holds:
Theorem 2.4.
[DJ12b] The set is spectral if and only if there exists a strongly continuous one parameter unitary group on with the property that, for all and :
(2.5) 
Moreover, given the unitary group , the spectrum of the selfadjoint infinitesimal generator of the group (as in Stone’s theorem), is a spectrum for .
Remark 2.5.
Example 2.6.
The simplest example of a spectral set is with spectrum . In this case, the group of local translations is
This can be checked by verifying that , , .
Proposition 2.7.
Let be a spectral set with spectrum . Let be a Lebesgue measurable subset of and such that . Then
(2.6) 
Proof.
Since is contained in , by Theorem 2.4, we have that, for almost every , . Thus, if , then for a.e., . On the other hand, since is unitary, we have that . But
so for a.e. .
∎
Next, we focus on spectral subsets of and define the oneparameter unitary group of local translations in an analogous way. As we will see, in this case, the parameter can be restricted from to and thus the unitary group of local translations is determined by a local translation unitary matrix.
Definition 2.8.
Let be a finite subset of . A group of local translations on is a continuous oneparameter unitary group , on with the property that
(2.7) 
A unitary matrix on is called a local translation matrix if
(2.8) 
If is a spectral subset of with spectrum and , we define the Fourier transform from to by the matrix:
(2.9) 
Let be the diagonal matrix with entries , . We define the group of local translations on associated to by
(2.10) 
The local translation matrix associated to is .
Proposition 2.9.
With the notations as in Definition 2.8,
(2.11) 
Thus the vectors in are the eigenvectors of corresponding to the eigenvalues of multiplicity one.
The matrix entries of are
(2.12) 
Proof.
We have , for all . The rest follows from an easy computation. ∎
Theorem 2.10.
Proof.
We have . Then . Hence . ∎
The converse holds also in the case of subsets of , i.e., the existence of a group of local translations, or of a local translation matrix guarantees that is spectral.
Theorem 2.11.
Let be a finite subset of . The following statements are equivalent:

is spectral.

There exists a unitary group of local translations , , on .

There exists a local translation matrix on .
The correspondence from (i) to (ii) is given by where is a spectrum for . The correspondence from (ii) to (iii) is given by . The correspondence from (iii) to (i) is given by: if is the spectrum of then is a spectrum for .
Proof.
The implications (i)(ii)(iii) were proved above. We focus on (iii)(i). Let be the spectrum of the unitary matrix , the eigenvalues repeated according to multiplicity and let be an orthonormal basis of corresponding eigenvectors. Let be the orthogonal projection onto . Then
We have, from (2.8),
so for all which implies that does not depend on , so it is equal to for some . Then for all and
so
Consider the matrices and . The previous equation implies that and since is unitary, we get that is also. But then the columns have unit norm so
and this implies that . The fact that the rows are orthonormal means that
But this means, first, that all the ’s are distinct and that is a spectrum for . ∎
Remark 2.12.
Given the group of local translations , , the local translation matrix is given by . Conversely, given the local translation matrix , this defines on in a unique way , . However, there are many ways to interpolate this to obtain a local translation group depending on the real parameter . One can pick some choices for such that is the spectrum of . Then consider the spectral decomposition
Define
Note that depends on the choice of . Any two such choices , are congruent modulo , and therefore the corresponding groups and coincide for .
Example 2.13.
The simplest example of a spectral set in is with spectrum . The local translation matrix associated to is the permutation matrix:
To see this, it is enough to check that, for ,
In the next proposition we link the two concepts for and for : if is a spectral set in , with spectrum , then is a spectral set in with spectrum . The local group of local translations for and can be expressed in terms of the local translation matrix associated to and .
Proposition 2.14.
Let be a spectral set in , with spectrum . Then the set is spectral in with spectrum . Define the matrix of the Fourier transform from to :
(2.13) 
and let be the diagonal matrix with entries , .
(2.14) 
The group of local translations associated to the spectrum of is given by
(2.15) 
where and represent the integer and the fractional parts respectively.
Proof.
The fact that is a spectrum for can be found, for example, in [KM06b]. The formula for appears in a slightly different form in [DJ12b], but we can check it here directly in a different way: it is enough to prove that
(2.16) 
Thus, we have to plug in in the right hand side of (2.15), with , . For the computation, we will use the following relation:
(2.17) 
Indeed, we have, for ,
because is a spectrum for .
In the following we present some connections between the local matrix and possible tilings for the set . We define a set as the set of powers of the matrix which have a canonical vector as a column, with 1 not on the diagonal.
Definition 2.15.
Let be a spectral subset of with spectrum . Let be the associated local translation matrix. Define
(2.19) 
Proposition 2.16.
Let be a spectral subset of with spectrum , . Assume . Assume in addition that the smallest lattice that contains is for some mutually prime integers . For a subset of the following statements are equivalent:

, in the sense that is a complete set of representatives modulo and every element in can be represented in a unique way as with and . In this case tiles by .

and .
Proof.
First, we present in a more explicit form. By Proposition 2.9 we have
Since we want to be 1 for some , we must have equality in the triangle inequality
and since this implies that so for all . Since the smallest lattice that contains is , we obtain that which means that . The converse also holds: if then has a 1 on position . Thus
(2.20) 
(i)(ii). Suppose there exists in such that . Then there exist in such that . Then , a contradiction. Also if , then .
(ii)(i). It is enough to prove that , because this implies that the map from to , is injective, and the condition implies that it has to be bijective.
Suppose not. Then there exist in in such that . Then . Therefore which contradicts the hypothesis. ∎
Corollary 2.17.
If the local translation matrix is
then tiles by .
Proof.
We have so one can take in Proposition 2.16. ∎
Another piece of information that is contained in the local translation matrix is the rationality of the spectrum:
Proposition 2.18.
Let be a spectral set in with spectrum and local translation matrix . Let , . Then if and only if . The spectrum is rational if and only if the group of local translations has an integer period, i.e., there exists , such that , .
Proof.
If , using equation (2.10), with , we have that so . Conversely, if then and therefore for all . The second statement follows from the first. ∎
3. Examples
In this section we study the local translation groups associated to spectral sets of low cardinality . Such sets were described in [DH12]. We recall here the results:
Definition 3.1.
The standard Hadamard matrix is
(3.1) 
We say that a matrix is equivalent to the standard Hadamard matrix if it can be obtained from it by permutations of rows and columns.
Let and be two subsets of and , . We say that is a Hadamard pair with scaling factor if is a spectrum for .
Theorem 3.2.
Let have elements and spectrum . Assume is in and . Suppose the Hadamard matrix associated to is equivalent to the standard by Hadamard matrix. Then has the form where is an integer and is a complete set of residues modulo with . In this case any such spectrum has the form where and are integers, is a complete set of residues modulo with greatest common divisor one, and where divides and is mutually prime with . The converse also holds.
Since for our Hadamard matrices are equivalent to the standard one (see [Haa97, TŻ06]) the next corollary follows:
Corollary 3.3.
A set with , , or , where is spectral if and only if where is a positive integer and is a complete set of residues modulo .
For cardinality the situation is more complex:
Theorem 3.4.
Let be spectral with spectrum and size . Assume is in both sets. Then there exists a set of integers , containing , and an integer scaling factor so that .
is a Hadamard pair (each containing ) of integers of size , with scaling factor , if and only if , , and , where and are all odd, is a positive integer, and are nonnegative integers, and divides , , , and , where is the greatest common divisor of the ’s and similarly for .
The next proposition helps us simplify our study:
Proposition 3.5.
Let be a spectral set in with spectrum , local translation group and local translation matrix . Let . Then is spectral with spectrum . The local translation group and the local translation matrix are related to the corresponding ones for and by
(3.2) 
Proof.
Everything follows from (2.10) by a simple calculation. ∎
N=2. We can take , with , and odd, and with odd. The matrix of the Fourier transform is
By equation (2.10), we can compute the local translation matrix and the local translation group :
(3.3) 
Here depends on the nonzero element of , called . Let , where is odd. Then , where is odd. Multiplying, we obtain
(3.4) 
We also have
(3.5) 
Note that when ,
In this case .
N=3. We can take and . The matrix of the Fourier transform is, with :
We compute the group of local translations:
(3.6) 
Multiplying, we obtain
Note that, when ,
In this case .
N=4. We take a simple case to obtain some nice symmetry, so we will ignore, after some rescaling, the common factor. So take , as in Theorem 3.4. The matrix of the Fourier transform is
(3.7) 
where . We compute integers powers of the spectral matrix :
(3.8) 
where .
We obtain for odd ,
(3.9) 
We obtain for even ,
(3.10) 
We compute . We have if and only if one of the following situations occurs: , , . This means that or