On Chebotarëv's nonvanishing minors theorem and the Biró-Meshulam-Tao discrete uncertainty principle

07/19/2018 ∙ by Stephan Ramon Garcia, et al. ∙ Pomona College California State University, Northridge 0

Chebotarëv's theorem says that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We then establish a generalization of the Biró-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions and twists. We then show that our result is best possible and always yields a lower bound at least as strong as Biró-Meshulam-Tao.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1. Introduction

Chebotarëv’s theorem says that every minor of a discrete Fourier matrix of prime order is nonzero; see [19, 4, 18, 9, 16, 8, 6, 20]. In 2005, Terence Tao provided a new proof of Chebotarëv’s theorem and obtained an improved uncertainty principle for complex-valued functions on prime fields [20]. This lower bound on the sum of the size of the support of a function and the size of the support of its Fourier transform was also independently discovered by András Biró [2] and Roy Meshulam [13] (see [8] and [20, p. 122] for details about the provenance of the result).

It is common to apply the Fourier transform to functions that exhibit some symmetry, for example, even or odd functions. We show that the lower bound in the Biró–Meshulam–Tao principle can be strengthened for these, and much more generally, for functions with symmetries arising from certain group actions. We prove broad generalizations of Chebotarëv’s theorem and the Biró–Meshulam–Tao principle, which yield uncertainty bounds that are best possible for the class of functions with the specified symmetry, and sometimes stronger than those provided by Biró–Meshulam–Tao. Moreover, our explorations in the case of non-prime fields reveal interesting phenomena that are worthy of further study (see Problem 5.13).

1.1. Nonvanishing minors and Chebotarëv’s theorem

A square matrix has the nonvanishing minors property if each minor of the matrix is nonzero. We do not restrict our attention to principal minors, that is, we permit the removal of any distinct rows and any distinct columns. We consider the determinant of the original matrix itself as one of its minors.

The matrix

(1)

in which , is the discrete Fourier transform matrix (or Fourier matrix) of order . It is symmetric, unitary, and satisfies .

If , in which , and if we index the rows and columns of from to , then the minor of that corresponds to rows and columns is zero since it is the determinant of the all-ones matrix. On the other hand, Chebotarëv’s theorem tells us that no minor of vanishes if is prime.

Theorem 1.1 (Chebotarëv).

has the nonvanishing minors property if and only if is prime or .

This was first posed to Chebotarëv by Ostrovskiĭ, who was unable to find a proof; see [19] for Chebotarëv’s proof and historical background. Chebotarëv’s theorem was independently rediscovered by Dieudonné in 1970 [4]. Other proofs can be found in [18, 4, 9, 16, 8, 7].

One of our main results (Theorem 4.8) is a broad generalization of Chebotarëv’s theorem that encompasses several other familiar matrices as special cases. We defer the general result, which is stated in terms of a general class of symmetries based on group actions, until Section 4.2 and instead devote the following section to a few special cases with commonly encountered symmetries. An exploration of the situation for non-prime fields is contained in Section 5.

1.2. Discrete cosine and sine transforms

For odd , the discrete cosine transform (DCT) matrix of modulus is the matrix with rows and columns indexed from to and whose entry in row and column is

In other words,

(2)

There are many variants of “the” discrete cosine transform matrix in the literature. The one selected above is natural from the perspective that it is real, symmetric, unitary, and satisfies . Discrete cosine transform matrices arise in many engineering and computer science applications, such as signal processing and image compression [10].

If is an odd composite number, we can write with . Then the minor of corresponding to rows and columns is zero. Thus, if has the nonvanishing minors property, then is not composite. The converse is also true.

Theorem 1.2.

Let be odd. The discrete cosine transform matrix has the nonvanishing minors property if and only if is prime or .

This result arises as a special case of a much more general theorem (Theorem 4.8) concerning Fourier analysis of functions that respect certain group actions; see Remark 4.9. In some instances, generalizations of Theorem 4.8 are possible over non-prime fields, although the details are subtle; see Section 5.

Theorem 4.8 also applies to the discrete sine transform matrix. For odd , the discrete sine transform (DST) matrix of modulus is the matrix with rows and columns indexed from to and whose entry in row and column is

In other words,

(3)

This matrix is real, symmetric, unitary, and satisfies . If is an odd composite number, we can write with . Then the -entry of is zero. Thus, must be prime for to have the nonvanishing minors property. The converse is also true.

Theorem 1.3.

Let be odd. The discrete sine transform matrix has the nonvanishing minors property if and only if is prime.

1.3. Uncertainty principles

Let be a prime and let be the field of order . Let denote the support of a function , that is, the subset of the domain of on which does not vanish. We use to denote the cardinality of a set. The Fourier transform of is the function defined by

(4)

In this context, the classical uncertainty principle states that

(5)

if [20, 5]. A remarkable improvement upon (5) is due, independently, to András Biró [2], Roy Meshulam [13], and Terence Tao [20] (see also [15, 14, 3]):

Theorem 1.4 (Biró–Meshulam–Tao).

If is not identically zero, then

(6)

The crucial improvement over (5) is the additive nature of (6). Theorem 1.4 is best possible in the following sense. Given with , there is an with and [20]. Chebotarëv’s theorem is at the heart of the proof.

The Biró–Meshulam–Tao uncertainty principle concerns generic functions from to . One might hope to obtain stronger versions for functions that enjoy certain symmetries. As a consequence of our generalized Chebotarëv theorem (Theorem 4.8) we obtain stronger versions of Theorem 1.4 for functions that respect certain group actions. Moreover, our lower bounds are never inferior to those of Biró–Meshulam–Tao. We require a bit of notation before presenting these results.

As before, let be a prime and let be the field of order . Let be a subgroup of the unit group (denoted ) and let be a character (a group homomorphism). A function such that for every and is called -symmetric. Some simple examples follow.

  • If , then is trivial and every function from to is -symmetric.

  • If is an odd prime, , and is the trivial character (the constant function on ), a -symmetric function is one with for all , that is, an even function.

  • If is an odd prime, , and is the character with , a -symmetric function is one with for all , that is, an odd function.

  • If , , and is the trivial character on , then a -symmetric function is one that is constant on each orbit in under the action of multiplication by elements of the subgroup . We call these orbits -orbits; they are the cosets of in and the singleton set . An -closed set is one that is a union of -orbits.

We have the following uncertainty principle for -symmetric functions, which is proved later as a special case of Theorem 6.5:

Theorem 1.5.

Let be a prime, let , and let be a character. Suppose that is a -symmetric function and .

  1. If is nontrivial, then

  2. If is trivial, then

Remark 1.6.

Since whenever admits a nontrivial character, our lower bounds are never worse than those of the Biró–Meshulam–Tao uncertainty principle (Theorem 1.4). We recover their result if and is the trivial character on .

The -symmetry of the function in Theorem 1.5 implies that the supports of both and are -closed (that is, unions of -orbits), and the orbit cannot be in the supports when is nontrivial. (See Lemma 3.3 and Corollary 3.9 for proofs.) Thus when precisely one of or vanishes at , we know that ; this can be combined with Theorem 1.4 to deduce the lower bound of given as the second case of Theorem 1.52. Similarly, when both and vanish at , we can deduce a lower bound of , which recapitulates Theorem 1.51, but this combination of Theorem 1.4 and careful counting is still strictly weaker than the result in the first case of Theorem 1.52.

We illustrate our uncertainty principle with some numerical examples.

Example 1.7.

If is an odd prime, is even, and , then

Following the counting considerations discussed in Remark 1.6, the support of an even function is even in size if vanishes at , or odd in size if does not vanish at , and the same principle applies to . Thus, when precisely one of or vanishes at , the sum of the sizes of their supports is odd, and so we can deduce the lower bound of from Theorem 1.4 and this counting principle. But the same technique applied to the case when both and vanish at cannot be used to improve the bound of given by Theorem 1.4. The results of this paper give the strictly stronger bound of .

Example 1.8.

Let and let have order . If is the trivial character on , then is -symmetric if and only if is constant on each of the -orbits

in . In particular, these -orbits reflect the multiplicative structure of rather than its additive structure. If is -symmetric, then

The lower bound of is what one obtains from Theorem 1.4. The lower bound of when precisely one of or vanishes at can be obtained from Theorem 1.4 if one recognizes that and modulo are and (not necessarily in that order) by the counting principle discussed in Remark 1.6. When both and vanish at , the same principle could be used to improve the lower bound of Theorem 1.4 to , but not to , which is given by the results of this paper.

Recall from Remark 1.6 that if is -symmetric for some character , then and are -closed (see Lemma 3.3 and Corollary 3.9). The following result, which is a special case of Theorem 6.9, shows that Theorem 1.5 is best possible.

Theorem 1.9.

Let be prime, let , and let be a character.

  1. If is nontrivial, then for any -closed subsets and of with

    there is a -symmetric with and .

  2. If is trivial and and are -closed subsets of with

    then there is a -symmetric with nd .

Tao [20] used the uncertainty theorem of Theorem 1.4 to obtain a novel proof of the Cauchy–Davenport theorem, a seminal result in additive combinatorics [21]. In some cases we can strengthen this theorem; see Section 6.4.

1.4. Organization of the paper

In Section 2 we establish some notation and review Fourier analysis on finite fields. In Section 3 we investigate -symmetry, which generalizes the underlying symmetry of the discrete cosine and sine transform matrices. In Section 4 we define a class of matrices for which a Chebotarëv-type theorem holds. We also study analogues for non-prime finite fields. In Section 5 we find (see Theorem 5.1) that if our group lies in a proper subfield, then the associated matrix does not have the nonvanishing minors property. This is always the case when in a non-prime field, so the analogues of the discrete cosine and sine transform matrices have vanishing minors. But we also find scenarios over non-prime fields that give rise to matrices with the nonvanishing minors property. We pose an open question (Problem 5.13) that asks for the precise condition needed to obtain the nonvanishing minors property over a general finite field. In Section 6 we prove our generalization (Theorem 6.5, which specializes to Theorem 1.5 above) of the Biró–Meshulam–Tao uncertainty principle. We also show that these results are best possible (in Theorem 6.9, which specializes to Theorem 1.9 above). We close with a discussion of the Cauchy–Davenport theorem.

2. Preliminaries

If and are sets, then denotes the set of all functions from into . If has a zero element and , then the support of is

(7)

The remainder of this section discusses the additive characters of finite fields and the discrete Fourier transform over finite fields that arises from them.

2.1. Finite fields and additive characters

Let denote the finite field of order . An additive character of is a group homomorphism from the additive group into the multiplicative group . The absolute trace from to its prime subfield is

The canonical additive character of is the function defined by

If is an additive character and , define by

(8)

Then is an additive character and . Thus, is the canonical additive character and is the trivial character, which maps everything to . Then

is the group of additive characters from into . The map is a group isomorphism from (under addition) to (under pointwise multiplication).

If , then

is a subset of that contains precisely characters. In particular, .

2.2. Group ring

Consider the group ring , whose elements we write as

(9)

We use brackets to distinguish elements of and when these have the same appearance (e.g., and ). If and are in , then , in which the coefficients

(10)

are obtained by convolution. Observe that is a -algebra that contains as an isomorphic copy of . One can regard each as a function by the formula . In this context, (7) suggests the definition

We apply an additive character to (9) by linear extension, that is,

(11)

2.3. Fourier transform

We shall require a more technical definition of (4) that works for all finite fields (not just those of prime order). The Fourier transform of is the function defined by

(12)

This induces an isomorphism

of -algebras, in which is equipped with pointwise multiplication. The inverse Fourier transform is defined by

The preceding definitions emphasize the difference between the operations on the domain (convolution) and codomain (pointwise multiplication). Some readers may prefer to use the same domain and codomain (regarded as vector spaces) with the different multiplications only implicitly acknowledged. We adopted this notation in Section

1.3 for the sake of simplicity. We offer the following translation between the two perspectives.

  • The domain of the Fourier transform can be regarded as rather than by applying the -vector space isomorphism that takes the group algebra element to the function with for every .

  • The codomain of the Fourier transform can be regarded as rather than by applying the -vector space isomorphism that takes to the function with for every .

Then the Fourier transform of is the function defined by

for every . If is the prime field , then

for every . This is the formula (4) from Section 1.3.

3. -symmetry

In this section we introduce the notion of -symmetry, which characterizes the functions used to form the discrete cosine matrix (2), discrete sine matrix (3), and their relatives. We then produce bases for the subspaces of -symmetric elements and their Fourier transforms. This permits us to define a general class of matrices that enjoy the nonvanishing minors property (Section 4).

3.1. Multiplication action

If , then acts on and on by multiplication:

The -orbit of is

If , then the preceding is the -coset in that contains . Consequently, the -orbits of are the -cosets that comprise the quotient group . The -orbits of are those of along with . An -closed subset of is one that is closed under the action of , that is, a union of -orbits.

We extend the action of to elements (9) of as follows:

(13)

The dot distinguishes this from the group ring product .

Similarly, acts on via

in which is defined by (8). The -orbits of are the sets for . Thus, the set of nontrivial characters is partitioned into orbits of characters each. The trivial character, , occupies its own orbit. An -closed subset of is one that is closed under the action of , that is, is a union of -orbits.

3.2. Characters of subgroups of and -symmetry

A character of is a group homomorphism . In particular, determines since the domain of a function is part of its definition. The set of all characters of is a group under pointwise multiplication. It is isomorphic to and contains the trivial character, which maps every element in to , as its identity element.

Suppose that and is a character. Then is -symmetric if

(14)

In light of (13), is -symmetric if and only if

For the rest of this paper, we use to denote the set of all -symmetric elements in when is a character of some subgroup of . The following is a consequence of commutativity and the distributive law in .

Lemma 3.1.

If and is a character of , then the set of all -symmetric elements in is a -vector subspace of .

This kind of symmetry is also respected by convolution in the following sense.

Lemma 3.2.

If and are characters from into , if is -symmetric, and if is -symmetric, then is -symmetric.

Proof.

For any ,

by (13)
by (10)
by (13)
by (14)
since

We next show that a -symmetric element of has a constrained support.

Lemma 3.3.

Let , let be a character, and let be -symmetric. Then is -closed and, if is nontrivial, .

Proof.

Since for all and for every , we see that is -closed. If is nontrivial, then there is an with . Consequently, and hence . ∎

We now consider some examples of -symmetry that encompass several familiar types of functions (e.g., even and odd functions).

Example 3.4.

If and is the trivial character, then every element of is -symmetric.

Example 3.5 (even element).

Suppose that is odd, , and is the trivial character. Then is -symmetric if and only if for every , that is, is even. Lemma 3.2 implies that the product of two even elements is even.

Example 3.6 (odd element).

Suppose that is odd, , and is the character of with . Then is -symmetric if and only if for every , that is, is odd. Moreover, Lemma 3.3 ensures since is nontrivial. The product of two odd elements is even by Lemma 3.2.

Example 3.7.

Suppose that and , in which is a primitive third root of unity in . Let be the character of with . Then is -symmetric if and only if for every . Since is nontrivial, Lemma 3.3 tells us that an element with this symmetry has .

3.3. Fourier characterization of -symmetry

We now show that -symmetry has a dual characterization in the Fourier domain.

Lemma 3.8 (Fourier characterization of -symmetry).

Let be a subgroup of and be a character. Then is -symmetric if and only if

(15)
Proof.

If , , and , then by (12), (8), and (11). If is -symmetric, then (14) and (12) ensure that the final expression becomes , thus proving (15). Conversely, if we assume (15), then the above calculation shows that for every and . Since and have the same Fourier transform for every , the invertibility of the Fourier transform implies that for every , that is, is -symmetric. ∎

We observe that -symmetry imposes constraints on the support of the Fourier transform of an element of . This is the Fourier analogue of Lemma 3.3.

Corollary 3.9.

Let , let be a character, and let be -symmetric. Then is -closed and, if is nontrivial, .

Proof.

Lemma 3.8 ensures that for and . Since , we see that is -closed. If is nontrivial, then there is an with . Consequently, , and hence . ∎

Corollary 3.10.

Let be a subgroup of , let be a character, and let denote the set of -symmetric elements in . Let be a set of representatives of the -orbits of (if is trivial) or of (if is nontrivial). If is -symmetric, then is uniquely determined by the values as runs through . That is, the map

(16)

from to is injective.

Proof.

Given , Corollary 3.9 enables us to reconstruct (apply the corollary if is nontrivial; already contains if is trivial). Since is a set of representatives of the -orbits of , Lemma 3.8 shows that the value for some determines for every . Thus, is determined for every ; that is, we can reconstruct the Fourier transform of . The invertibility of the Fourier transform ensures that we can reconstruct . ∎

In fact, the map (16) is bijective; this is Proposition 3.17 below.

3.4. Basis for the space of -symmetric elements

Let and let be a character. For each , define

(17)

These are convenient -symmetric elements that we shall use to construct certain matrices later on.

Example 3.11.

Suppose that is the trivial group and is the trivial character. Then for each .

Example 3.12.

Let be odd, , and be the trivial character. For each , we have , which is even in the sense of Example 3.5.

Example 3.13.

Let be odd, , and be the character with . For each , we have , which is odd in the sense of Example 3.6. In particular, .

Example 3.14.

Let and , in which is a primitive third root of unity in . If is the character with , then

for each . In particular, ; see Example 3.7.

The following lemma explains the properties of the that we have observed.

Lemma 3.15.

Let , let be a character, and let . Then

  1. is -symmetric;

  2. if is trivial or ;

  3. if is nontrivial.

Proof.

If and , then is -symmetric since

by (17) and (13)

If , then the summands in (17) run through a set of nonzero -scalar multiples of the elements of . Consequently, . On the other hand . If is trivial, this is , whose support is . If is nontrivial, then and hence . ∎

The fact that distinct -orbits are disjoint leads to the following conclusion.

Corollary 3.16.

Let , let be a character, and let be a set of representatives of the -orbits of (if is trivial) or of (if is nontrivial). Then is a -linearly independent subset of the -vector subspace of -symmetric elements of .

In fact, we can prove a much stronger result.

Proposition 3.17.

Let , let be a character, and let be the set of -symmetric elements in . Let be sets of representatives of the -orbits of (if is trivial) or of (if is nontrivial). Then is a -basis of (which is -dimensional) and the map from to is a -vector space isomorphism.

Proof.

Recall from Lemma 3.1 that is -vector space and consider the maps

in which is the inclusion map (valid by Corollary 3.16) and the second map is , which Corollary 3.10 ensures is injective. Both maps are injective, so the -dimension of the vector spaces involved does not decrease. However, by Corollary 3.16 and . Thus, all three spaces have dimension and hence both maps are -vector space isomorphisms. Since is linearly independent (Corollary 3.16) and spans , it is a basis of . ∎

3.5. Basis for the space of Fourier transforms of -symmetric elements

We now introduce a natural basis for , the space of Fourier transforms of -symmetric elements of . If , then we define