Fractal Uncertainty Principles for Ellipsephic Sets
51
Fractal Uncertainty Principles for Ellipsephic Sets by Nicholas Hu B.Sc. (Hons.), The University of British Columbia, A T S P F R D M S in The Faculty of Graduate and Postdoctoral Studies (Mathematics) T U B C (Vancouver) April © Nicholas Hu,
Transcript of Fractal Uncertainty Principles for Ellipsephic Sets
Fractal Uncertainty Principlesfor Ellipsephic Sets
by
Nicholas Hu
B.Sc. (Hons.), The University of British Columbia, 2019
A Thesis Submitted in Partial Fulfillment of
the Reqirements for the Degree of
Master of Science
in
The Faculty of Graduate and Postdoctoral Studies
(Mathematics)
The University of British Columbia
(Vancouver)
April 2021
© Nicholas Hu, 2021
The following individuals certify that they have read, and recommend to
the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis
entitled
Fractal Uncertainty Principles for Ellipsephic Sets
submitted by Nicholas Hu in partial ful�llment of the requirements for the
degree of Master of Science in Mathematics.
Examining Committee:
Izabella Łaba, Mathematics, UBC
Co-supervisor
Malabika Pramanik, Mathematics, UBC
Co-supervisor
ii
Abstract
Chicken chicken chicken (CCC) chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken Chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken, chicken Chicken-chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken chicken, chicken chicken chicken chicken chicken Chicken chicken Chicken chicken chicken chicken chicken chicken chicken-chicken chicken chicken CCC chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken, chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken, chicken chicken chicken chicken chicken chicken, chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken chicken, chicken chicken chicken chicken chicken chicken chicken chicken-chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken. Chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken chicken-chicken chicken chicken chicken chicken CCC chicken.Fractal uncertainty principles (FUPs) in harmonic analysis quantify the
extent to which a function and its Fourier transform can be simultaneously
localized near a fractal set. We investigate the formulation of such princi-
ples for ellipsephic sets, discrete Cantor-like sets consisting of integers in a
given base with digits in a speci�ed alphabet. We employ a combination
of theoretical and numerical methods to �nd and support our results.
To wit, we resolve a conjecture of Dyatlov and Jin by constructing a
sequence of base-alphabet pairs whose FUP exponents converge to the
basic exponent and whose dimensions converge to δ for any given δ ∈ ( 12, 1),
thereby con�rming that the improvement over the basic exponent may
be arbitrarily small for all δ ∈ (0, 1). Furthermore, using the theory of
prolate matrices, we show that the exponents β1 of the same sequence
decay subexponentially in the base.
In addition, we explore extensions of our work to higher-order el-
lipsephic sets using blocking strategies and tensor power approximations.
We also discuss the connection between discrete spectral sets and base-
alphabet pairs achieving the maximal FUP exponent.
iii
Lay summary
The mathematical �eld of harmonic analysis studies transformations of
one object into another and the relationships between the original object
and the transformed one. Generally speaking, the more “certain” one is
about the properties of the original object, the less “certain” one can be
about the properties of the transformed object, and vice-versa. Mathemati-
cal explanations of this phenomenon are accordingly called uncertainty
principles.
Recently, there has been interest in formulating such principles for
objects associated with fractal sets, which can be thought of as patterns
of shapes containing smaller copies of themselves. In our research, we
investigated ellipsephic sets – a particular kind of fractal set – and found
new principles along with evidence in support of hypothesized ones.
iv
Preface
This thesis consists of original, unpublished, and independent work by the
author. Some of this work was expository in nature; for such work, the
source of the expounded material was cited and the presentation of the
material was substantially modi�ed so as to constitute a novel exposition.
Namely, Chapters 2 and 4 and Section 3.1 comprise known results,
whereas Sections 3.2 to 3.4 comprise new results. Our main theoretical
�ndings are Proposition 3.2 and Theorem 3.4 and their corollaries.
v
Table of contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Lay summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of �gures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Fractal uncertainty principles . . . . . . . . . . . . . . . . . . . . 32.1 Continuous fractal uncertainty principles . . . . . . . . . . . 3
2.2 Discrete fractal uncertainty principles . . . . . . . . . . . . . 6
2.2.1 Elementary properties . . . . . . . . . . . . . . . . . 7
2.2.2 Improvements over the basic exponent . . . . . . . 10
3 Upper bounds on exponents . . . . . . . . . . . . . . . . . . . . . 133.1 Small alphabets . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Large alphabets . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Prolate matrix bounds . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 First-order compressions . . . . . . . . . . . . . . . . 21
3.3.2 Higher-order compressions . . . . . . . . . . . . . . 27
3.4 Tensor power approximations . . . . . . . . . . . . . . . . . . 30
vi
4 Maximal exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Spectral alphabets . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Quasi-spectral alphabets . . . . . . . . . . . . . . . . . . . . . 34
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 40
vii
List of tables
3.1 Subexponential decay of β1 for large-alphabet sequences . . . . 21
5.1 Comparison of bounds on β − β0 . . . . . . . . . . . . . . . . . . 36
viii
List of �gures
2.1 FUP exponents for alphabets with M ≤ 10 . . . . . . . . . . . . 11
2.2 Close-up of Figure 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 FUP exponents for large-alphabet sequences . . . . . . . . . . . 18
3.2 Subexponential decay of β1 for large-alphabet sequences . . . . 20
3.3 Tensor power approximation of an ellipsephic compression . . 32
ix
Acknowledgements
First and foremost, I would like to extend my deepest gratitude to my
supervisors, Professors Izabella Łaba and Malabika Pramanik. I will for-
ever be indebted to them for their insightful teaching, patient advising,
assiduous mentorship, and perspicacious guidance.
Furthermore, I would like to express my appreciation to Professors
Uri Ascher and Chen Greif for introducing me to scienti�c research and
training me in numerical analysis.
I would also like to convey my gratefulness to Mike Gelbart for showing
me the path that led me here and to Ed Knorr for facilitating my journey
on it.
Finally, I would like to thank Jenally and JeongSoo for their unfaltering
emotional support. This work would not have been possible without them.
x
I will give thanks to you, O Lord, among the peoples;
I will sing praises to you among the nations.
For your steadfast love is as high as the heavens;
your faithfulness extends to the clouds.
Psalm 57:9–10
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1 Introduction
Fractal uncertainty principles (FUPs) are manifestations of the eponymous
harmonic-analytic paradigm for functions supported near fractal sets. They
were �rst introduced and employed by Dyatlov and Zahl in their study of
spectral gaps and have since found numerous applications to problems in
quantum chaos [DZ16; Dya19]. Such principles can be formulated in both
continuous and discrete settings, although our focus will be on the latter.
The version of the uncertainty principle whereupon continuous FUPs
are based is as follows. For a small positive parameter h, let Fh : L2(R) →L2(R) be the unitary “semiclassical” Fourier transform operator given by
Fh f (ξ) :� 1√2πh
∫R
e−ixξ/h f (x) dx.
De�nition 1.1. Two h-dependent families of sets {Xh}, {Yh} ⊆ P(R)are said to satisfy an uncertainty principle with exponent β ∈ R if
‖1XhFh1Yh
‖L2(R)→L2(R) � O(hβ) as h → 0.
Thus, according to this incarnation of the uncertainty principle, if f is
supported in Yh , then Xh cannot contain more than O(hβ) of the L2mass
of Fh f .
Note. For the sake of conciseness, we will elide the subscripts of X and
Y in the sequel.
Replacing the continuous Fourier transform with its discrete counter-
part yields a natural analogue of De�nition 1.1. To wit, let FN : `2(ZN ) →`2(ZN ) be the unitary discrete Fourier transform operator given by
FN u( j) :� 1√N
∑`∈ZN
e−2πi j`/N u(`),
where ZN :� Z/NZ � {0, 1, . . . ,N − 1}.
1
De�nition 1.2. Two N-dependent families of sets {XN }, {YN } ⊆ P(ZN )are said to satisfy an uncertainty principle with exponent β ∈ R if
‖1XNFN1YN
‖`2(ZN )→`2(ZN )
� O(N−β) as N →∞.
Note. In this context, we will frequently (and tacitly) regard the operator
above as an N ×N matrix and thence the `2(ZN )-`2(ZN ) norm as a matrix
2-norm, with the ambient vector norm being ‖u‖`2(ZN ):� (∑ j∈ZN
|u( j)|2)1/2.We will also elide the subscripts of X and Y as in the continuous setting.
When X and Y are “fractal” in senses that will be delineated in Chap-
ter 2, the uncertainty principles that they satisfy are called fractal uncer-
tainty principles. In the same chapter, we will show that families of both
continuous and discrete fractal sets trivially satisfy uncertainty principles
with an exponent depending on the “dimension” of the fractal sets. This
naively obtained exponent is termed the basic FUP exponent.
From these preliminary observations, two cardinal questions arise: can
the basic FUP exponent be improved upon, and, if so, what is the best
possible exponent achievable by two given families of fractal sets? The
former question was answered a�rmatively (in nondegenerate cases) by
Bourgain, Dyatlov, and Jin [BD18; DJ18; DJ17]; the latter is what we will
address ourselves to in the present work. Speci�cally, we will examine
families of ellipsephic sets, which are a type of discrete fractal set.
In Chapter 3, we will investigate the bounding of β from above and
prove the main results of this thesis: Proposition 3.2 and Corollary 3.2.1,
which show that the improvement over the basic exponent can be arbitrar-
ily small within a certain range of dimensions, and Theorem 3.4, which
gives a new upper bound on a quantity related to β.
Finally, in Chapter 4, we will discuss the conditions under which the
best possible β is maximal among fractal sets of a given dimension and
adduce numerical evidence for an existing conjecture in this regard.
2
2 Fractal uncertainty principles
In this chapter, we will �rst give a brief introduction to fractal uncertainty
principles for the continuous Fourier transform – which is to say, in their
original and more widely used setting – before presenting their analogues
for the discrete Fourier transform. Although a detailed discussion of the
former is beyond the scope of this work, we will prove some “basic” esti-
mates whose counterparts are equally fundamental in the latter context.
In the discrete setting, we will also show how these basic estimates can
be improved upon, contextualizing the main topics and �ndings of our
research.
2.1 Continuous fractal uncertainty principles
Let us begin by recalling De�nition 1.1 – that two h-dependent families
of sets {Xh}, {Yh} ⊆ P(R) satisfy an uncertainty principle with exponent
β ∈ R if
‖1XFh1Y ‖L2(R)→L2(R) � O(hβ) as h → 0.
Example 2.1. Let X � Y � [0, h]. By Hölder’s inequality,
‖1XFh1Y ‖L2→L2 ≤ ‖1[0,h]‖L∞→L2 ‖Fh ‖L1→L∞ ‖1[0,h]‖L2→L1
� h1/2 · (2πh)−1/2 · h1/2 ,
so X and Y satisfy an uncertainty principle with exponent1
2.
Our interest is in the case where X and Y are “fractal”, which we take
to mean Ahlfors-David regular in the undermentioned sense.
De�nition 2.2. Suppose that X ⊆ R is nonempty and closed, δ ∈ [0, 1],CR ∈ [1,∞), and αmin , αmax ∈ [0,∞] with αmin ≤ αmax. The set X is said
to be δ-regular with constant CR on scales αmin to αmax if there exists a
3
2.1. Continuous fractal uncertainty principles
locally �nite measure µX supported on X such that
1
CR|I |δ ≤ µX(I) ≤ CR |I |δ
for every interval I with length |I | ∈ [αmin , αmax] centred at a point in X.
The exponent δ is then called the dimension of X.
Note. When speaking of such sets, we will take “interval” to mean
“closed interval”.
Remark. If I is an interval (not necessarily centred at a point in X) and
Imin (resp. Imax) is the interval of length |I | centred at min(I ∩ X) (resp.
max(I∩X)), then I∩X ⊆ Imin∪Imax. Hence µX(I) ≤ 2CR |I |δ . For this reason,
the centring condition on I for the upper bound on µX(I) is sometimes
omitted from this de�nition. Be this as it may, the precise value of CR is
typically irrelevant for our purposes.
Example 2.3.
(a) The set {0} is 0-regular with constant 1 on scales 0 to ∞.
(b) The set R is 1-regular with constant 1 on scales 0 to ∞.
(c) The set [0, α], α > 0 is 0-regular with constant 2 on scales α to ∞.
(Roughly speaking, [0, α] “looks like” a point at these scales.)
(d) The set [0, α], α > 0 is 1-regular with constant 2 on scales 0 to α.
(Roughly speaking, [0, α] “looks like” a line at these scales.)
Now, given δ and CR, we seek the largest exponent β such that an
uncertainty principle with exponent β is satis�ed by all families of sets
X,Y ⊆ [0, 1] that are δ-regular with constant CR on scales h to 1.
To begin with, we observe that
‖1XFh1Y ‖L2(R)→L2(R) ≤ 1
since Fh is unitary. On the other hand, it can be shown that the volume
(that is, the Lebesgue measure) of such sets is bounded above by h1−δ,
whence we obtain
‖1XFh1Y ‖L2(R)→L2(R) . h1
2−δ
4
2.1. Continuous fractal uncertainty principles
by arguing as in Example 2.1. The volume bound is in turn an immediate
consequence of the following property (taking ρ � h and I � [0, 1]).
Proposition 2.4 (Lemma 2.8, [BD18]: The small cover property). Let
X ⊆ R be δ-regular with constant CR on scales αmin to αmax. If ρ ≥ αmin
(with ρ , 0) and I is an interval with |I | ∈ [ρ, αmax], then there exists a
coverJof I ∩X consisting of nonoverlapping intervals of length ρ such that
#J≤ 24C2
R(|I |/ρ)δ. (We say that two intervals are nonoverlapping if their
intersection contains at most one point.)
Proof. Let J be the cover of I ∩ X consisting of all intervals of the
form ρ[ j, j + 1], j ∈ Z that intersect I ∩ X. For each J ∈ J, let J′ ⊃ J be
the concentric interval of length 2ρ. Since J intersects X, the interval J′
contains one of length ρ centred at a point in X. Hence
µX(J′) ≥ C−1R ρδ .
Now each point in⋃
J∈J J′ lies in at most 3 of the J′, and⋃
J∈J J′ ⊆ I+Bρ+ρ/2can be covered by 4 intervals of length |I | since |I + B3ρ/2 | � |I | + 3ρ ≤ 4|I |.It follows that
#J · C−1R ρδ ≤∑J∈J
µX(J′) ≤ 3µX
(⋃J∈J
J′)≤ 12 · 2CR |I |δ .
Together, these estimates furnish a lower bound for β known as the
basic fractal uncertainty principle (FUP) exponent:
β0 :� max
{0,1
2− δ
}.
Although it is impossible to improve upon this exponent using only the
volumes of X and Y, improvements can be made for all δ ∈ (0, 1) by
exploiting their fractal structure [BD18; DJ18].
5
2.2. Discrete fractal uncertainty principles
2.2 Discrete fractal uncertainty principles
The discrete “fractal” sets that will be the object of our study are ellipsephic
sets, otherwise known as discrete Cantor sets in the work of Dyatlov et al.∗
De�nition 2.5. An ellipsephic† set in base M is a set consisting of all
k-digit integers in base M with digits in some nonempty alphabetA⊆ ZM .
Such a set is denoted Ck(M,A) (or simply Ck in unambiguous contexts)
and k ∈ N is called the order of the set. In other words,
Ck � Ck(M,A) :�{
k−1∑d�0
ad Md: ad ∈ A
}.
Note. Hereinafter, we will adopt Dyatlov and Jin’s use of | · | for cardi-
nality, having no further occasion to use it for measure.
From this de�nition, it is clear that Ck ⊆ ZN for N :� Mkand that
|Ck | � |A|k � N logM |A| . Furthermore, consideration of the measure µCk:�∑
c∈Ckδc shows that Ck is (logM |A|)-regular (in the sense of De�nition 2.2)
on scales 1 to N , motivating the next de�nition.
De�nition 2.6. The dimension of Ck(M,A) is δ :� logM |A|.
Remark. This dimension coincides with the Hausdor� dimension of the
Cantor set C(M,A) :� ⋂k∈N
⋃j∈Ck (M,A)[ j, j + 1]/Mk ⊆ [0, 1]. For instance,
C(3, {0, 2}) is the (log32)-dimensional ternary Cantor set.
Evidently, δ ∈ (0, 1) unless |A| � 1 or |A| � M. We will see shortly that
uncertainty principles for such alphabets are trivial, so we dispense with
their consideration without loss of generality (which, in e�ect, allows us
to assume that M ≥ 3).
Now, given M and A, we seek uncertainty principles for X � Y �
Ck(M,A) regarded as subsets of ZN (as k ranges over N). This gives a
∗We have elected to use their preexisting and more descriptive name, attributed to Mauduit
by Col [Col06].
†This word is a translation of the French ellipséphique, which Mauduit coined as a port-
manteau of the Greek words ἔλλειψις (élleipsis, “ellipsis”) and ψηφίο (psifío, “digit”). We
prescribe the English pronunciation [I.lIp"sEf.Ik].
6
2.2.1. Elementary properties
sequence of values of N for which the statement in De�nition 1.2 amounts
to
‖1CkFN1Ck
‖2 .M,A N−β .
Once again, we have
‖1XFN1Y ‖2 ≤ 1
since FN is unitary. On the other hand, we obtain the analogue of the
volume-related bound much more easily:
‖1XFN1Y ‖2 ≤ ‖1XFN1Y ‖F �
√|X | |Y |
(1√N
)2� Nδ− 1
2 ,
where ‖·‖F denotes the Frobenius norm (also known as the Hilbert–Schmidt
norm). Thus we derive the same lower bound for β:
β0 � max
{0,1
2− δ
}.
As before, a greater value of β cannot be achieved by considering the
cardinality of Ck alone. Nonetheless, as Dyatlov and Jin showed, one can
leverage its fractal structure to realize an improvement for any nontrivial
dimension δ [DJ17]; we will present their argument in Section 2.2.2.
2.2.1 Elementary properties
Before discussing improvements over the basic exponent, we will compile
some properties of the norms rk � rk(M,A) :� ‖1Ck (M,A)FN1Ck (M,A)‖2 that
will be employed throughout this work.
The �rst is a lower bound on rk that translates into an upper bound on
β that matches β0 when δ � 0 or δ � 1, vindicating our ostensible neglect
of those cases.
Proposition 2.7. Let X,Y ⊆ ZN . Then
‖1XFN1Y ‖2 ≥√
max{|X |, |Y |}N
.
7
2.2.1. Elementary properties
Proof. If y ∈ Y, then 1{y} is a unit vector with ‖(1XFN1Y)1{y}‖2 �√|X |/N; similarly, if x ∈ X, then 1{x} is a unit vector with ‖(1XFN1Y)∗
1{x}‖2 �√|Y |/N .
Corollary 2.7.1. If X � Y � Ck , then
β ≤ 1 − δ2
.
Note. Hereinafter, ωN will denote the Nthroot of unity e−2πi/N
.
Proposition 2.8. Let X,Y ⊆ ZN . If nX , nY ∈ ZN , then
‖1X+nXFN1Y+nY
‖2 � ‖1XFN1Y ‖2 .
Proof. Let LN denote the left shift operator on `2(ZN ) so that 1X+nX�
L−nXN 1X LnX
N . Since FN MN � LNFN , where MN is the unitary multiplication
operator MN u( j) :� ω jN u( j) on `2(ZN ), the norm above is invariant under
shifting of X; the proof for shifts of Y is similar.
In view of the corollary hereunder, we may restrict our attention to
alphabets containing 0 without any loss of generality.
Corollary 2.8.1. If a ∈ ZM and A⊆ {0, 1, . . . , (M − 1) − a}, then
rk(M,A+ a) � rk(M,A).
Proof. Apply the preceding proposition to X � Y � Ck(M,A) with
nX � nY �∑k−1
d�0 aMd. (The latter hypothesis ensures that there is no
“carryover” in the addition.)
The results proved heretofore have not used the fractal structure of
ellipsephic sets in any way. By contrast, it is central to the next result,
which depends on the fact that if k � k1 + k2, each base-M numeral in
Ck is a numeral in Ck1concatenated with a numeral in Ck2
. This enables
an underlying DFT of size N � Mkto be meaningfully decomposed into
two DFTs of sizes N1 � Mk1 and N2 � Mk2 (as in the Cooley–Tukey FFT
algorithm), yielding the following submultiplicative property.
Lemma 2.9 (Lemma 4.6, [Dya19]). If k1 , k2 ∈ N, then rk1+k2≤ rk1
rk2.
8
2.2.1. Elementary properties
Proof. Let k :� k1 + k2 and N j :� Mk j . We will view rk as the norm of
the operator Fk :� 1CkFN restricted to `2Ck
:� {u ∈ `2(ZN ) : supp u ⊆ Ck}.Given u ∈ `2Ck
and v :� Fk u, let U and V be the |A|k1 × |A|k2 matrices
formed by reshaping the Ck-indexed entries of u and v in column- and row-
major order, respectively. (Here we use the fact that Ck � N1Ck2+ Ck1
�
N2Ck1+ Ck2
.) By construction, ‖U‖F � ‖u‖2, ‖V ‖F � ‖v‖2, and
Vab �1√N
∑p∈Ck
1
q∈Ck2
ω(N2a+b)(N1q+p)N Upq (a ∈ Ck1
, b ∈ Ck2).
Since N1N2 � N , we have
Vab �1√N
∑p∈Ck
1
q∈Ck2
ωapN1
ωbpN ω
bqN2
Upq
�1√N1
∑p∈Ck
1
ωapN1
ωbpN
©«1√N2
∑q∈Ck
2
ωbqN2
Upq
ª®®¬ .
Thus, the matrix V can be obtained from U via the following sequence of
transformations:
(1) Taking the DFT of each row of U, obtaining the matrix U′ :� UFËk2
.
(2) Multiplying the entries of U′ by twiddle factors, obtaining the matrix
V′ with V′pb :� ωbpN U′pb .
(3) Taking the DFT of each column of V′, obtaining the matrix V :�
Fk1V′.
Consequently,
‖v‖2 � ‖V ‖F≤ ‖Fk1
‖2‖V′‖F (3)
� ‖Fk1‖2‖U′‖F (2)
≤ ‖Fk1‖2‖Fk2
‖2‖U‖F (1)
� rk1rk2‖u‖2.
9
2.2.2. Improvements over the basic exponent
2.2.2 Improvements over the basic exponent
We are now in a position to establish the aforementioned improvement
over the basic FUP exponent as stated in the theorem below.
Theorem 2.10. Suppose that Ck(M,A) is an ellipsephic set of dimension
δ ∈ (0, 1). Then there exists a β > β0 (depending only on M and A) for which
‖1CkFN1Ck
‖2 .M,A N−β .
By Lemma 2.9, the sequence {logM rk} is subadditive, which allows us
to invoke the lemma below from real analysis.
Lemma 2.11 (Fekete’s lemma). Suppose that {an} ⊆ [−∞,∞) is a subad-
ditive sequence (that is, an1+n2≤ an1
+ an2for all n1 , n2). Then limn→∞
ann �
infn∈Nann .
To wit, it su�ces to show that rk < min{1,Nδ−1/2} for a single k to
prove Theorem 2.10, which is accomplished by the lemmata hereunder.
Lemma 2.12 (Lemma 4.7, [Dya19]). If δ < 1, then rk < 1 for all su�-
ciently large k.
Proof. The basic FUP gives rk ≤ 1. If rk � 1, there must exist a nonzero
u with ‖1CkFN1Ck
u‖2 � ‖u‖2. Hence both u and FN u must be supported
in Ck . Now FN u( j) � N−1/2p(ω jN ), where p is the polynomial p(z) :�∑N−1
`�0 u(`)z` . We also note that |A| ≤ M − 1 since δ < 1, and in particular,
that ZM \A, �.
By de�nition, Ck excludes the numbers aMk−1through (a + 1)Mk−1 − 1,
where a ∈ ZM \ A. By Proposition 2.8, we may cyclically shift Ck so as
to assume that a � M − 1, which implies that deg p < (M − 1)Mk−1since
supp u ⊆ Ck .
On the other hand, since suppFN u ⊆ Ck , the polynomial p has at
least N − |Ck | roots, and this number is at least Mk − (M − 1)k . But for all
su�ciently large k, this exceeds the upper bound on deg p, necessitating
rk < 1.
Lemma 2.13 (Lemma 4.8, [Dya19]). If δ > 0, then rk < Nδ−1/2 for k ≥ 2.
10
2.2.2. Improvements over the basic exponent
Proof. The basic FUP gives rk ≤ Nδ−1/2. Suppose for the sake of con-
tradiction that k ≥ 2 yet rk � Nδ−1/2. Otherwise stated, the spectral and
Frobenius norms of 1CkFN1Ck
are equal, so this operator must have rank
1. In particular, all its 2 × 2 minors vanish; that is,
det
[ω
j`N ω
j`′
N
ωj′`N ω
j′`′
N
]� ω
j`+ j′`′
N − ω j`′+ j′`N � 0 for all j, j′, `, `′ ∈ Ck ,
which implies that ( j − j′)(` − `′) ∈ NZ. However, since δ > 0, we have
|A| > 1, so there exist distinct j, j′ ∈ Ck with | j − j′ | < M � N1/k ≤ N1/2,
whence a contradiction is obtained by taking ` � j and `′ � j′.
Having found that the basic FUP exponent can be improved upon, it is
natural to ask how the best possible exponent β for a given base-alphabet
pair (M,A) depends on M and A. As Figure 2.1 shows, the quantitative
nature of this dependence – which is the topic of our research – does not
appear to be simple.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
δ
β
M � 3
M � 4
M � 5
M � 6
M � 7
M � 8
M � 9
M � 10
Figure 2.1. Numerically approximated FUP exponents for all nontrivial
alphabets with M ≤ 10.
Notably, the improvement over the basic exponent (plotted in blue)
appears to be quite small for certain bases and alphabets, especially for
11
2.2.2. Improvements over the basic exponent
alphabets with dimensions greater than1
2; see Figure 2.2. The qualitative
question of whether this gap can be arbitrarily small and the quantitative
question of how small this gap can be are the subject of Chapter 3.
Speci�cally, Proposition 3.2 and Corollary 3.2.1 establish that the di�er-
ence can indeed be arbitrarily small in the range δ ∈ ( 12, 1), complementing
the results of Dyatlov and Jin that demonstrate the same for δ ∈ (0, 12]
(Proposition 3.1 and Corollary 3.1.1). Our Theorem 3.4 further shows that a
quantity related to β can be subexponentially small within the same range
δ ∈ ( 12, 1), roughly corresponding to the fact that some points in this region
of the plot are extremely close to the blue line (but see Figure 3.2 for a
more precise exhibition).
0.5 0.6 0.7 0.8 0.9 1.010−15
10−10
10−5
100
δ
β
M � 3
M � 4
M � 5
M � 6
M � 7
M � 8
M � 9
M � 10
Figure 2.2. Log-linear close-up of Figure 2.1 in the range δ ≥ 1
2.
We also see that the maximal value of β, given by Corollary 2.7.1
(plotted in red), is sometimes attained. Su�cient and potentially necessary
conditions for this to occur are discussed in Chapter 4.
12
3 Upper bounds on exponents
Although the basic FUP exponent can be improved upon for any δ ∈ (0, 1),the gain is marginal for some bases and alphabets. Indeed, Dyatlov and
Jin showed that for any δ ≤ 1
2, there exists a sequence of base-alphabet
pairs whose dimensions converge to δ and whose (best) FUP exponents
converge to β0, and they conjectured the same for δ > 1
2[DJ17; Dya19].
More precisely, let δ(M,A) :� logM |A| and βk(M,A) :� − logN rk(M,A)(N � Mk
) so that the best FUP exponent for Ck(M,A) is
β(M,A) :� limk→∞
βk(M,A) � sup
k∈N
βk(M,A)
by Lemma 2.11. Dyatlov and Jin established upper bounds on β that
entail the existence of a sequence {(M j ,Aj)} with δ(M j ,Aj) → δ and
β(M j ,Aj) → β0 for any given δ ≤ 1
2. We expound their result and tech-
niques in Section 3.1 and then extend them to the regime δ > 1
2in Section 3.2,
thereby a�rming their conjecture. (Accordingly, we secern alphabets with
δ ≤ 1
2from those with δ > 1
2by terming the former “small” and the latter
“large”.) Following this, we attempt to enhance our quantitative estimate
on the rate of convergence of β to β0, which also prompts us to explore
how β might be approximated by βk more generally.
Note. The glyph C and its subscripted forms will denote positive abso-
lute constants. The unsubscripted form will represent a “generic” constant
that may assume di�erent values in each of its instances (for example,
C + C � C).
13
3.1. Small alphabets
3.1 Small alphabets
To obtain upper bounds on β, it will prove useful to study the DFT of the
indicator function of Ck :
FN1Ck( j) � 1√
N
∑`∈ZN
ωj`N 1Ck
(`)
�1√N
∑a0 ,...,ak−1∈A
k−1∏d�0
ωjad Md
N
�
k−1∏d�0
1√M
∑ad∈A
ωjad Md
N
�
k−1∏d�0
1√M
∑a∈A
exp(−2πi jaMd−k)
�
k∏s�1
TA
(j
Ms
), (3.1)
where we have introduced the 1-periodic function
TA(x) :�1√M
∑a∈A
exp(−2πiax) for x ∈ R.
We will call this function the trigonometric mask polynomial of the al-
phabet A since the polynomial PA(x) �∑
a∈A xais known as the mask
polynomial of A in the study of spectral and tiling sets [LLR13; Mal20].
Proposition 3.1 (Proposition 3.17, [DJ17]). Suppose that 2 ≤ A ≤√
M
and let A :� {0, 1, . . . ,A − 1} ⊆ ZM . Then
β(M,A) ≤ 1
2− δ + CA2
M logM.
We pause to outline the main strategy of the proof, as this strategy will
be reused in the next section.
To bound β from above, we will bound rk from below by applying the
operator 1CkFN1Ck
to 1Ck. Working with r2k for simplicity, we will obtain
the lower bound ‖FN1Ck‖2`2(Ck )
/Ak. We will then bound ‖FN1Ck
‖2`2(Ck )
from below by identifying a subset of digits B ⊆ A such that TA – and
hence FN1Ck– is large on Ck(M,B) ⊆ Ck(M,A).
14
3.1. Small alphabets
Since the trigonometric mask polynomial of the alphabet above is large
near integers, we will aim to make the fractional parts of j/Mssmall for all
j ∈ Ck(M,B) and s ∈ N. As the number j/Msis simply the number j with
its base-M radix point shifted left s places, we will choose the smallest
digits in A for B.
Proof. Recognizing the trigonometric mask polynomial of Aas a partial
sum of a geometric series, we compute that
TA(x) �1√M· 1 − exp(−2πixA)1 − exp(−2πix)
(x < Z),
and TA(x) � A/√
M if x ∈ Z. Noting that |1−exp(−2πiθ)| � 2 |sin(πθ)| for
θ ∈ R, we obtain
|TA(x)| �1√M
����sin(πxA)sin(πx)
���� (x < Z).
One can verify that |TA| is decreasing on [0, 1
2A ], which implies that
|TA(x)| ≥A
C0
√M
for all x ∈ [0, 1
2A ]
and hence for all x ∈ R with frac(x) ∈ [0, 1
2A ] since TA is 1-periodic (here
we can take C0 :� π/2).
Now let B :� {0, 1, . . . , bA/4c} ⊆ A. For all j ∈ Ck(M,B) and s ∈ N,
we have
0 ≤ frac
(j
Ms
)≤
s−1∑r�0
br Mr−s(br ∈ B)
≤ A4· 1
M − 1≤ A
4· 1
A2 − 1since A ≤
√M
≤ 1
2Asince A ≥ 2.
Thus, in view of Equation (3.1),
|FN1Ck (M,A)( j)|2 ≥(
A
C0
√M
)2k
for all j ∈ Ck(M,B).
15
3.1. Small alphabets
It follows that
r2k ≥‖1Ck
FN1Ck‖2`2
‖1Ck‖2`2
�1
Ak
∑j∈Ck (M,A)
|FN1Ck (M,A)( j)|2
≥ 1
Ak
∑j∈Ck (M,B)
|FN1Ck (M,A)( j)|2
≥(|B|A
) k (A
C0
√M
)2k
≥ A2k
(2C0)2k Mk,
which gives
β ≤ 1
2− δ + log(2C0)
logM
≤ 1
2− δ + C1A2
M logM(C1 :� log(2C0)/C2),
provided that A2 ≥ C2M.
If A2 < C2M, we use the fact that
TA(x) �A√M(1 + O(Ax)) for all x ∈ [0, 1
2A ],
where |O(Ax)| ≤ C3Ax. Indeed, for all j ∈ Ck(M,A) and s ∈ N, we have
0 ≤ frac
(j
Ms
)<
AM − 1
≤ 2AM
≤ 1
2A
if we take C2 ≤ 1/4, inasmuch as A2 < C2M. As a result,����TA
(j
Ms
)���� ≥ A√M
(1 − C4A2
M
)(C4 :� 2C3),
16
3.2. Large alphabets
whence we may conclude that
β ≤ 1
2− δ − log(1 − C4A2/M)
logM
≤ 1
2− δ + C5A2
M logM(C5 :� log(4)C4)
after choosing C2 ≤ 1/(2C4).
Corollary 3.1.1. For any δ ∈ (0, 12], there exists a sequence {(M j ,Aj)} of
base-alphabet pairs such that δ(M j ,Aj) → δ and β(M j ,Aj) → 1
2− δ � β0.
Proof. For each j ∈ N, take A � j + 1 and M ≈ A1/δabove.
3.2 Large alphabets
Having seen the proof of Proposition 3.1, it is natural to ask whether it
can be modi�ed to produce a comparable bound for large alphabets. If we
use the same alphabet and constrain the base to be of size ≈ A1/δfrom the
outset, then choosing B judiciously – by selecting the smallest ≈ A1/δ−1
digits of A– does in fact accomplish this.
Proposition 3.2. For any δ ∈ ( 12, 1) and A ≥ 2, let M :� 4bA1/δc and
A :� {0, 1, . . . ,A − 1} ⊆ ZM . Then
β(M,A) . 1
logM.
Proof. By construction, we have |A| � A, and A ⊆ ZM since M ≥A1/δ > A. Moreover, the trigonometric mask polynomial of A is given by
the same expression as in Proposition 3.1.
Let B :� {0, 1, . . . , bA1/δ−1c}, which is a subset of Asince δ ∈ ( 12, 1) and
A ≥ 2. Arguing as in the proof of Proposition 3.1, we obtain
0 ≤ frac
(j
Ms
)≤ bA
1/δ−1cM − 1
for all j ∈ Ck(M,B) and s ∈ N.
Now the upper bound is again at most1
2A , since
2AbA1/δ−1c ≤ 2bA1/δc ≤ 4bA1/δc − 1.
17
3.2. Large alphabets
Thus,
|FN1Ck (M,A)( j)|2 ≥(
A
C0
√M
)2k
for all j ∈ Ck(M,B).
This yields the bound
r2k ≥(|B|A
) k (A
C0
√M
)2k
≥ Ak/δ
C02k Mk
≥ 1
(2C0)2k,
whence
β ≤ log(2C0)logM
.
Corollary 3.2.1 (Conjecture 4.4, [Dya19]). For any δ ∈ ( 12, 1), there exists
a sequence {(M j ,Aj)} of base-alphabet pairs such that δ(M j ,Aj) → δ and
β(M j ,Aj) → 0 � β0.
Proof. For each j ∈ N, take A � j + 1 above, noting that δ(M j ,Aj) → δ
since M ≈ A1/δ.
101.0
101.2
101.4
101.6
101.8
10−5
10−4
10−3
10−2
10−1
100
M
β
δ � 0.55δ � 0.60δ � 0.65δ � 0.70δ � 0.75δ � 0.80δ � 0.85δ � 0.90δ � 0.95
Figure 3.1. FUP exponents for some sequences given by the proof of
Corollary 3.2.1.
18
3.3. Prolate matrix bounds
Examining Figure 3.1, it seems plausible that the proof of Corollary 3.2.1
actually gives sequences with β polynomially small in M.
Conjecture 3.3. For any δ ∈ ( 12, 1), there exists a sequence {(M j ,Aj)} of
base-alphabet pairs such that δ(M j ,Aj) → δ with β polynomially small in
M; that is,
β(M j ,Aj) .1
Mcj
for some c > 0 depending on δ.
In the subsequent sections of this chapter, we will present some results
and observations that are potentially related to our conjecture.
3.3 Prolate matrix bounds
The numerical experiments that have generated the �gures thus far –
Figures 2.1, 2.2, and 3.1 – have all approximated the FUP exponent β(M,A) �supk∈N βk(M,A) by βk(M,A) for small values of k in light of the fact that βk
“appears almost constant for k ≥ 3” [DJ17] (see Appendix A for details). On
the basis of such experiments, Dyatlov suggests that there might even be
a sequence of base-alphabet pairs with β exponentially small in M [Dya19].
If this approximation is reasonable, the following result lends credence to
Dyatlov’s suggestion and a fortiori our own Conjecture 3.3.
Theorem 3.4. For any δ ∈ ( 12, 1), suppose that 2 ≤ A < M with M ≈ A1/δ ,
and let A :� {0, 1, . . . ,A − 1} ⊂ ZM . Then
β1(M,A) .δ1
logMexp
(−M2δ−1
logM
).
Corollary 3.4.1. For any δ ∈ ( 12, 1), there exists a sequence {(M j ,Aj)} of
base-alphabet pairs such that δ(M j ,Aj) → δ with β1 subexponentially small
in M.
Proof of corollary. For each j ∈ N, take A � j + 1 and M � 4bA1/δcabove (as in the proofs of Proposition 3.2 and its corollary).
In Figure 3.2, we plot the quantities γ1 :� − log(β1 logM) against M for
some sequences given by the proof of Corollary 3.4.1. As expected, the γ1
19
3.3. Prolate matrix bounds
exhibit subpolynomial growth, corresponding to subexponential decay of
the β1.
101
102
103
104
105
10−0.5
100.0
100.5
101.0
101.5
M
γ1
δ � 0.55δ � 0.60δ � 0.65δ � 0.70δ � 0.75δ � 0.80δ � 0.85δ � 0.90δ � 0.95
Figure 3.2. γ1 :� − log(β1 logM) for some sequences given by the proof of
Corollary 3.4.1.
Furthermore, after performing linear regressions to determine approx-
imate powers c for which γ1 ∼ Mc, we obtain values that are only slightly
larger than those for the upper bound of Theorem 3.4, suggesting that the
bound is nearly tight (see Table 3.1).
To prove the theorem, we will consider the Gram matrix of the A × A
leading principal submatrix FM of 1AFM1A and show that its largest
eigenvalues – and hence r1 � ‖FM ‖2 – are very close to unity for large A.
Our argument relies on the fact that there exists a matrix with the same
eigenvalues as F∗MFM that is well-approximated by a so-called prolate
matrix. This matrix is in turn well-approximated by a projection matrix
and thus links a matrix with known eigenvalues to the one we wish to
study (see Lemmas 3.5 and 3.6).
We will also employ the following terminology to facilitate our forth-
coming discussion. Recall that the operator 1CkFN1Ck
regarded as an
operator from `2(Ck) to itself is called the compression of FN to `2(Ck),
20
3.3.1. First-order compressions
δ 2δ − 1 c
0.55 0.1 0.110
0.60 0.2 0.220
0.65 0.3 0.317
0.70 0.4 0.422
0.75 0.5 0.533
0.80 0.6 0.613
0.85 0.7 0.730
0.90 0.8 0.830
0.95 0.9 0.984
Table 3.1. Approximate powers c for which γ1 ∼ Mcfor some sequences
given by the proof of Corollary 3.4.1.
while the natural number k is called the order of the ellipsephic set Ck .
We will therefore call such an operator a kth-order (ellipsephic) compres-sion. In other words, Theorem 3.4, which we will prove in the upcoming
subsection, is derived from a (lower) bound on the norms of �rst-order
compressions.
Note. For greater concordance with the literature, we will use zero-
based indexing for vectors and matrices throughout this section. In addi-
tion, we will suspend the practice of using “N” to denote Mkin the next
subsection to allow it to denote the size of a prolate matrix.
In a similar vein, σi(A) will denote the (i + 1)th largest singular value
of a matrix A and λi(A) will denote the (i + 1)th largest eigenvalue of a
Hermitian matrix A.
3.3.1 First-order compressions
Suppose that N ∈ N and W ∈ (0, 12). The N × N prolate matrix BN,W is
the symmetric Toeplitz matrix whose entries are
BN,W (m , n) :�sin(2πW(m − n))
π(m − n) ,
where it is understood that BN,W (m , n) � 2W for m � n [Kar+19; Var93].
This matrix arose from a series of papers published between 1961 and
1978 by Landau, Pollak, and Slepian, who considered the extent to which
21
3.3.1. First-order compressions
a bandlimited signal could also be approximately timelimited. This led
to their introduction of prolate spheroidal wave functions (PSWFs) in the
continuous case and discrete prolate spheroidal sequences (DPSSs) in the
discrete case [Kar+19; Sle78].
By construction, the DPSSs are N discrete-time sequences whose
DTFTs are bandlimited to [−W,W] for some W ∈ (0, 12) and concentrated at
the temporal indices {0, 1, . . . ,N−1}. It can be shown that the �nite-length
vectors (sometimes called Slepian basis vectors) formed by restricting these
sequences to {0, 1, . . . ,N−1} are an orthonormal eigenbasis for the prolate
matrix [Kar+19].
Now consider the A ×A matrix FM formed by removing the zero rows
and columns of 1AFM1A (with A, M, and Aas in Theorem 3.4), that is to
say, the A × A leading principal submatrix of 1AFM1A. We observe that
F∗MFM(m , n) �
1
M
A−1∑k�0
(ωn−mM )k
�1
M· 1 − z−2A
1 − z−2, z :� eπi(n−m)/M
�1
M· z
zA ·zA − z−A
z − z−1
� z1−A · sin(π(n − m)A/M)M sin(π(n − m)/M) ,
so F∗MFM is unitarily similar to the N × N matrix BM,N,W de�ned by
BM,N,W (m , n) :�sin(2πW(m − n))
M sin(π(m − n)/M)
with N � A < M and W �A2M ∈ (0, 12 ) (via the change of basis matrix
exp(−πi · diag(0, 1, . . . ,A − 1) · (1 − A)/M)).The apparent resemblance between BM,N,W and BN,W is concretized by
the forthcoming Lemma 3.5, which expresses the former as an additive
perturbation of the latter. The subsequent Lemma 3.6 shows that BN,W can
in turn be approximated by an orthogonal projection in the same manner.
To wit, each perturbation is the sum of a low-rank and a small-norm matrix,
so the largest eigenvalues of BM,N,W – and hence F∗MFM – must be near
unity as a result of Weyl’s inequality. The degree of their propinquity is,
in essence, the content of Theorem 3.4.
22
3.3.1. First-order compressions
Lemma 3.5. Let M ∈ N and suppose that N < M and W ∈ (0, 12). Then
for any ε > 0, there exist symmetric matrices L, E ∈ RN×N such that
BM,N,W � BN,W + L + E
with
rank(L) . max
{− log(ε(α2 − 1))
log(α2), 1
}(α :� M/N) and ‖E‖ . ε.
Proof. We follow the argument used by Zhu et al. in the proof of their
theorem on the spectral concentration of BM,N,W [Zhu+18, Theorem 1].
Using the Laurent series
csc(x) �∞∑
k�0
(−1)k+12(22k−1 − 1)B2k
(2k)! x2k−1(x ∈ (−π, π))
�
∞∑k�0
2(1 − 2−(2k−1))ζ(2k)π2k
x2k−1
evaluated at x :� π(m − n)/M, we can write
(BM,N,W − BN,W )(m , n) �sin(2πW(m − n))
M
(csc(x) − 1
x
)� LK(m , n) + EK(m , n),
where
LK(m , n) :�sin(2πW(m − n))
M
K∑k�1
ck x2k−1 ,
EK(m , n) :�sin(2πW(m − n))
M
∞∑k�K+1
ck x2k−1 ,
ck :�2(1 − 2−(2k−1))ζ(2k)
π2k.
Clearly, LK , EK ∈ RN×N
are symmetric. Now
LK(m , n) �sin(2πWm) cos(2πWn) − cos(2πWm) sin(2πWn)
M· pK(m , n),
23
3.3.1. First-order compressions
where pK is a bivariate polynomial of degree at most 2K−1 in each of m and
n. Hence, if d j,` is the coe�cient of m j n` in pK and SK , CK ∈ RN×2K ,UK ∈
R2K×2K
are the matrices de�ned by
SK(m , j) :� sin(2πWm) · m j ,
CK(n , j) :� cos(2πWn) · n j ,
UK( j, `) :� d j,` ,
then
LK(m , n) �2K−1∑j,`�0
SK(m , j)UK( j, `)CK(n , `) − CK(m , j)UK( j, `)SK(n , `)
� (SKUKCËK − CKUKSËK)(m , n).
It follows that rank(LK) ≤ 2K + 2K � 4K.
As for the matrix EK , we have the entrywise bound
|EK(m , n)| ≤1
M
∞∑k�K+1
�����2(1 − 2−(2k−1))ζ(2k)π2k
(π(m − n)
M
)2k−1�����
.1
M
∞∑k�K+1
(NM
)2k−1since lim
k→∞ζ(2k) � 1
�1
N· α−2K
α2 − 1α :� M/N
≤ εN
provided that
K ≥ − log(ε(α2 − 1))
log(α2)�: K(ε).
For such K, the Gershgorin circle theorem∗
implies that
‖EK ‖ � ρ(EK) ≤ maxm
∑n,m
|EK(m , n)| . ε
∗The Gershgorin circle theorem states that each eigenvalue of a (complex) square matrix
A �: [ai j] lies in some Gershgorin disc Bri(aii), ri :�
∑j,i |ai j | [HJ13, Theorem 6.1.1].
24
3.3.1. First-order compressions
since EK(m ,m) � 0 for all m. The conclusion therefore holds with L � LK
and E � EK for K � max{dK(ε)e , 0}.
The asymptotic behaviour of the eigenvalues of the prolate matrix was
�rst determined by Slepian, who showed that the �rst 2NW eigenvalues of
BN,W tend to unity (exponentially quickly) for large N , while the rest tend
to zero [Sle78]. Recently, Karnik et al. derived non-asymptotic bounds for
its eigenvalues by relating BN,W to the Gram matrix of low-frequency DFT
vectors [Kar+19]. Namely, if FN,W ∈ CN×(2bNWc+1)
is the matrix consisting
of the length-N DFT vectors with frequencies between −bNWc/N and
+bNWc/N , that is
FN,W ( j, `) :�1√Nω
j(`−bNWc)N ,
then BN,W is related to FN,W F∗N,W ∈ RN×N
as follows.
Lemma 3.6 (Theorem 1, [Kar+19]). Suppose that N ∈ N and W ∈ (0, 12).
Then for any ε ∈ (0, 12), there exist symmetric matrices L, E ∈ R
N×N such
that
BN,W � FN,W F∗N,W + L + E
with
rank(L) � O(log(N) log(1/ε)) and ‖E‖ ≤ ε
as N →∞ and ε→ 0.
Proof. See Karnik et al. [Kar+19].
Finally, we conjoin the approximations of Lemmas 3.5 and 3.6 to pro-
duce an approximation of BM,N,W by the projection FN,W F∗N,W .
Proof of Theorem 3.4. Suppose that ε ∈ (0, 12). By combining Lem-
mas 3.5 and 3.6 with N � A < M and W �A2M ∈ (0, 12 ), we obtain
BM,A,A/2M � FA,A/2MF∗A,A/2M + L + E
25
3.3.1. First-order compressions
for some symmetric matrices L, E ∈ RA×A
with rank(L) ≤ `(M,A, ε) and
‖E‖ . ε, where
`(M,A, ε) :� Cmax
{− log(ε(α2 − 1))
log(α2), 1
}+ O(log(A) log(1/ε))
(α :� M/A; A→∞, ε→ 0).
(For the sake of brevity, we will omit the matrix subscripts for N and W in
what follows.)
Now let f (M,A) :� rank(FF∗) � 2bA2/2Mc + 1. Then by Weyl’s in-
equality†,
λ f−1−`(F∗MFM) � λ f−1−`(BM)
≥ λ f−1(FF∗ + E) + λN−1−`(L)
≥ λ f−1(FF∗ + E) since rank(L) ≤ `
≥ λ f−1(FF∗) − Cε since ‖E‖ . ε
� 1 − Cε since FF∗ is a projection.
With a view to estimating λ0(F∗MFM), let ε � ε(M,A) be the minimal
ε > 0 for which f − 1 − ` ≥ 0. On the one hand, we see that f − 1 ≈ A2−1/δ
(where δ > 1
2). On the other hand, we have
− log(ε(α2 − 1))log(α2)
�log(1/ε)log(α2)
− log(α2 − 1)log(α2)
� Oδ
(log(1/ε)log(A)
)+ Oδ(1)
since α2 ≈ A2(1/δ−1)and δ < 1, so the second term of ` is dominant for
large A. It follows that
ε .δ exp
(−A2−1/δ
logA
).
†Weyl’s inequality states that if A and B are n × n Hermitian matrices, then λi(A + B) ≥λ j(A) + λn−1−( j−i)(B) for 0 ≤ i ≤ j ≤ n − 1 [HJ13, Theorem 4.3.1].
26
3.3.2. Higher-order compressions
Thus, for all su�ciently large A, we have r21 � λ0(F∗MFM) ≥ 1 − Cε,
whence log r1 &δ −ε. We conclude that
β1 � − logM r1 .δε
logM
.δ1
logMexp
(−M2δ−1
logM
).
3.3.2 Higher-order compressions
We will now attempt to generalize the results of the previous subsection
to higher-order compressions. To this end, suppose that k ∈ N and let FN
be the Ak × Akmatrix formed by removing the zero rows and columns of
1CkFN1Ck
. For p , q ∈ Ck , we compute that
F∗N FN (p , q) �
1
N
∑r∈Ck
ω(q−p)rN
�1
N
∑a0 ,...,ak−1∈A
k−1∏d�0
ω(q−p)ad Md
N
�1√N
k∏s�1
TA
(q − pMs
),
following the calculation at the beginning of this chapter. Recalling from
the proof of Proposition 3.1 that
TA(x) �1√M· 1 − exp(−2πixA)1 − exp(−2πix)
�1√M· z1−A · sin(πxA)
sin(πx) , z :� eπix ,
we obtain
F∗N FN (p , q) �
k∏s�1
z1−As ·
sin(π(q − p)A/Ms)M sin(π(q − p)/Ms)
, zs :� eπi(q−p)/Ms.
Hence F∗N FN is unitarily similar to the Ak×Ak
matrix B(k)M,A,W whose entries
are given by
B(k)M,A,W (p , q) :�k∏
s�1
sin(2πW(p − q)/Ms−1)M sin(π(p − q)/Ms)
,
27
3.3.2. Higher-order compressions
which we will seek to express as a perturbation of some matrix B(k)M,A,W to
be speci�ed later.
Let us exemplify a possible generalization in the second-order case. The
A2 × A2matrix B(2)M,A,W is block symmetric Toeplitz with A × A symmetric
Toeplitz blocks; that is,
B(2)M,A,W �:
T0,0 · · · T0,A−1...
...
TA−1,0 · · · TA−1,A−1
, Ta1 ,b1∈ R
A×A(a1 , b1 ∈ A)
with Ta1 ,b1� Tb1 ,a1
, Ta1+1,b1+1� Ta1 ,b1
, and Ta1 ,b1symmetric Toeplitz. If the
rows and columns of each block are indexed by A, then
Ta1 ,b1(a0 , b0) � B(2)M,A,W ((a1a0)M , (b1b0)M)
�sin(2πW1(p − q)) sin(2πW2(p − q))
M2csc(x1) csc(x2),
where p :� (a1a0)M , q :� (b1b0)M , Ws :� A/2Ms, and xs :� π(p − q)/Ms
.
Now |x1−n1π | < π for n1:� a1− b1 since |(a0)M −(b0)M | < A; similarly,
|x2 | < π since |(a1a0)M − (b1b0)M | < A2. Hence, for any �xed a1 and b1, we
have
csc(x1) � (−1)n1∞∑
k�0
ck(x1 − n1π)2k−1 ,
csc(x2) �∞∑
k�0
ck x2k−12 ,
where the coe�cients ck are as in the proof of Lemma 3.5. We then de�ne
the corresponding block Ta1 ,b1of B(2)M,A,W such that subtracting it from Ta1 ,b1
cancels the principal parts of these Laurent series; to wit,
Ta1 ,b1(a0 , b0) :�
sin(2πW1(p − q)) sin(2πW2(p − q))M2
·[(−1)n1
x1 − n1π· 1
x2+
(−1)n1x1 − n1π
(csc(x2) −
1
x2
)+
(csc(x1) −
(−1)n1x1 − n1π
)1
x2
].
28
3.3.2. Higher-order compressions
From here, we may separate each series into lower- and higher-order
terms as in the �rst-order case, although cross-terms now arise from
multiplying them together:
Ta1 ,b1− Ta1 ,b1
� LK + E(1,2)K + E(2,1)K + E(2,2)K �: LK + EK ,
where
LK(a0 , b0) :�sin(2πW1(p − q)) sin(2πW2(p − q))
M2· Σ(1)K (x1)Σ
(2)K (x2),
E(1,2)K (a0 , b0) :�sin(2πW1(p − q)) sin(2πW2(p − q))
M2· Σ(1)K (x1)Σ
(2)>K(x2),
E(2,1)K (a0 , b0) :�sin(2πW1(p − q)) sin(2πW2(p − q))
M2· Σ(1)>K(x1)Σ
(2)K (x2),
E(2,2)K (a0 , b0) :�sin(2πW1(p − q)) sin(2πW2(p − q))
M2· Σ(1)>K(x1)Σ
(2)>K(x2);
Σ(1)K (x1) :� (−1)
n1K∑
k�1
ck(x1 − n1π)2k−1 ,
Σ(1)>K(x1) :� (−1)
n1∞∑
k�K+1
ck(x1 − n1π)2k−1 ,
and likewise for Σ(2)K (x2) and Σ
(2)>K(x2).
By the same arguments as in the proof of Lemma 3.5, we obtain
rank(LK) ≤ 22(2(2K − 1) + 1) ≤ 16K and���� 1M · Σ(1)>K(x1)
���� . 1
A· α−2K
α2 − 1,
���� 1M · Σ(2)>K(x2)���� . 1
A· α−4K
α4 − 1.
To bound the cross-terms, we employ the similarly derived estimates���� 1M · Σ(1)K (x1)���� . 1
A· 1
α2 − 1,
���� 1M · Σ(2)K (x2)���� . 1
A· 1
α4 − 1.
In fact, for our purposes, it su�ces to use α−2Kin place of α−4K
and α2 − 1in place of α4 − 1. The resulting entrywise bound is
|EK(a0 , b0)| .1
A2· α−2K
(α2 − 1)2≤ ε
A2,
29
3.4. Tensor power approximations
provided that
K ≥ − log(ε(α2 − 1)2)
log(α2)�: K(2)(ε).
Finally, since these estimates hold for an arbitrary block of B(2)M,A,W −B(2)M,A,W (partitioned as above), we arrive at the following analogue of
Lemma 3.5.
Lemma 3.7. Let M ∈ N and suppose that A < M and W ∈ (0, 12). Then
for any ε > 0, there exist real symmetric matrices L, E such that
B(2)M,A,W � B(2)M,A,W + L + E
with
rank(L) . A2 ·max{K(2)(ε), 1} and ‖E‖ . ε,
where
K(2)(ε) :� − log(ε(α2 − 1)2)
log(α2)(α :� M/A).
However, we were unable to devise a congruous generalization of
Lemma 3.6, which would seem to require specifying a basis for the image
of a suitable low-rank projection.
3.4 Tensor power approximations
Returning to the problem of relating β to βk for general bases and alphabets,
suppose βk were approximately constant in that βk ≈ β1. Expressed in
terms of norms, this would mean that rk ≈ rk1 . As submultiplicativity
(Lemma 2.9) implies that rk ≤ rk1 , any increase in βk may be reckoned to
the failure of the reverse inequality.
Consequently, it is natural to compare the operator FN , whose norm is
rk by de�nition, to the operator F⊗kM , whose norm is rk
1 . Indeed, the phase
of the (a , b)-entry of FN (as indexed by a , b ∈ Ck) is
−2πN(ak−1Mk−1
+ · · · + a0M0)(bk−1Mk−1+ · · · + b0M0)
≡ −2π(
ak−1b0 + · · · + a0bk−1M
+ · · · +a0b0Mk
)(mod 2π),
30
3.4. Tensor power approximations
while that of F⊗kM is
−2π(
ak−1bk−1 + · · · + a0b0M
).
Thus, after permuting the columns of the tensor power such that the
base-M expansions of their indices are colexicographically ordered, the
phases match to the �rst order (in M−1). Even so, it can be shown that the
contribution of the higher-order terms vitiates entrywise approximation
of FN by F⊗kM .
Nevertheless, numerical experiments such as the one depicted in Fig-
ure 3.3 suggest that not only is the operator norm of F⊗kM close to that
of FN , but that every singular value of F⊗kM is close to the corresponding
singular value of FN . Although we have yet to �nd an explanation of this
phenomenon, such an explanation might lead to a useful relation between
β1 and βk . For instance, if the behaviour of the relative error εk in the
approximation ‖FN ‖ ≈ ‖F⊗kM ‖ were known, one could compute βk via the
formula
βk(M,A) � β1(M,A) +η(M,A)logM
,
where
η(M,A) :� sup
k∈N
log(1 + εk(M,A))k
.
31
3.4. Tensor power approximations
0 50 100 150 200 250
10−10
10−8
10−6
10−4
10−2
100
i
σi(FN )σi(F⊗k
M )
Figure 3.3. Singular values of an ellipsephic compression and its tensor
power approximation (k � 4).
32
4 Maximal exponents
In contrast to the bases and alphabets hitherto examined – whose FUP
exponents are nearly minimal – there exist bases and alphabets that achieve
the maximal FUP exponent1−δ2
. As we will see, alphabets that are a certain
type of discrete spectral set always possess this property. Conversely, all
alphabets attaining the maximal exponent satisfy a condition that is slightly
weaker than the one de�ning those spectral sets, although numerical
experiments suggest that the di�erence is merely nominal (for small bases,
at least) [DJ17]. In this chapter, we provide a brief exposition of these facts.
4.1 Spectral alphabets
For each ` ∈ ZM , let e` be the discrete exponential function
e`( j) :� FM1{`}( j) �1√Mω
j`M .
De�nition 4.1. A nonempty set A⊆ ZM is said to be a (discrete) spec-tral setmodulo M if there exists a set B ⊆ ZM equinumerous to A– called
a spectrum for A– such that {eb}b∈B is `2(A)-orthogonal.
To this standard de�nition, we append another requirement that will
be of import to us.
De�nition 4.2. A spectral alphabet in base M is an alphabet in base
M that is a spectral set modulo M and a spectrum for itself.
Example 4.3. Let p be prime and q ∈ N be indivisible by p. Then
A :� qZp is a spectral alphabet in base M :� pq.
To see this, let b , b′ ∈ Awith b , b′, and write b � qc , b′ � qc′ for some
c , c′ ∈ Zp with c , c′. Then
〈eb , eb′〉`2(A) �1
M
∑a∈A
ωa(b−b′)M �
1
M
∑r∈Zp
ωqr(c−c′)p � 0,
33
4.2. �asi-spectral alphabets
provided that p - q(c − c′), which is guaranteed by Euclid’s lemma.
We refer the reader to Dyatlov and Jin for additional examples of
spectral alphabets [DJ17]. Within the context of discrete FUPs, the interest
in such alphabets stems from the following property that they possess.
Proposition 4.4 (Proposition 3.13, [DJ17]). If A ⊆ ZM is a spectral al-
phabet, then rk(M,A) � (|A|/M)k/2, whence
β(M,A) � 1 − δ2
.
Proof. Let u ∈ `2(ZM). Then
‖1AFM1Au‖2`2(ZM )
�
1AFM
(∑a∈Au(a)1{a}
) 2`2(ZM )
� ∑
a∈Au(a)ea
2`2(A)
�∑
a∈A|u(a)|2‖ea ‖2`2(A) since A is spectral
�|A|M‖1Au‖2
`2(ZM )
≤ |A|M‖u‖2
`2(ZM ),
so r1 ≤ (|A|/M)1/2. Submultiplicativity (Lemma 2.9) then yields an upper
bound on rk that is matched from below by Proposition 2.7.
Remark. For such alphabets, the square of the largest singular value
of 1CkFN1Ck
(that is, r2k ) is (|A|/M)k � Nδ−1. On the other hand, this
operator has at most |Ck | � Nδnonzero singular values, and the sum of
their squares is ‖1CkFN1Ck
‖2F� N2δ−1
. Hence 1CkFN1Ck
must have Nδ
nonzero singular values, all equal to Nδ−1.
4.2 Quasi-spectral alphabets
Recall from Section 3.1 that the trigonometric mask polynomial of an
alphabet A⊆ ZM is
TA(x) �1√M
∑a∈A
exp(−2πiax).
34
4.2. �asi-spectral alphabets
An alphabet is therefore spectral in base M if and only if
TA
(b − b′
M
)� 0 for all distinct b , b′ ∈ A.
We accordingly give the following name to a formally weaker condition
that has remained unnamed to date in the literature.
De�nition 4.5. A quasi-spectral alphabet in base M is an alphabet in
base M such that
TA
(b − b′
M
)TA
(b − b′
M2
)� 0 for all distinct b , b′ ∈ A.
Clearly, all spectral alphabets are quasi-spectral alphabets. If the con-
verse were true, the following proposition would be the converse of Propo-
sition 4.4. In fact, Dyatlov and Jin veri�ed computationally that the spectral
and quasi-spectral alphabets for M ≤ 25 are identical, and we continued
this veri�cation up to M ≤ 39. We share their belief that this is true in
general, despite being unable to demonstrate this.
Proposition 4.6 (Proposition 3.16, [DJ17]). If A⊆ ZM and
β(M,A) � 1 − δ2
,
then A is a quasi-spectral alphabet.
Proof. See Dyatlov and Jin [DJ17].
35
5 Conclusion
As a result of our work and that of Dyatlov and Jin, we now possess a
rudimentary understanding of fractal uncertainty principles for ellipsephic
sets: for nontrivial alphabets, an improvement over the basic exponent
can always be achieved (Theorem 2.10), but for any given dimension, it
can always be arbitrarily small (Corollaries 3.1.1 and 3.2.1).
Quantitatively speaking, lower bounds on the di�erence between
β(M,A) and β0(M,A) are also known and are compared in Table 5.1 to
the upper bounds of Chapter 3.
Lower bound Upper bound
Small alphabets
(δ ≤ 1
2)
M−8+o(1)
[DJ17, Cor. 3.5]
M−(1−2δ)+o(1)
(Proposition 3.1)
Large alphabets
(δ > 1
2)
exp(−CMδ/(1−δ)+o(1))[DJ17, Cor. 3.7]
(logM)−1
(Proposition 3.2)
Table 5.1. Comparison of lower and upper bounds on β − β0.
While the upper bound of Proposition 3.2 is far from the corresponding
lower bound, Theorem 3.4 suggests that it might be possible to narrow
this gap considerably. Future work that does so could involve further
investigation of the strategies in Sections 3.3.2 and 3.4, or even the study of
di�erent alphabets altogether (that is, other than the �rst A digits of ZM).
In this regard, it is likely that matrix analysis will again prove e�ectual.
We are also optimistic that forthcoming advances in the theory of
spectral sets will enable us to show that quasi-spectrality is equivalent
to spectrality for alphabets and hence characterize those attaining the
maximal exponent.
Moreover, there remain several largely unexplored ways of generalizing
ellipsephic-set FUPs themselves. One extension replaces the DFT operator
36
with the “dilated” DFT operator
FN,αu( j) :� 1√N
∑`∈ZN
ωj`N,αu(`),
where ωN,α :� e−2πiα/Nand α ∈ [1,M]. Dyatlov and Jin observed ex-
perimentally that for many values of α (other than 1), these operators
admit uncertainty exponents depending only on the dimension of the set;
whether this is true more generally remains an open question. Another gen-
eralization considers multidimensional DFTs and ellipsephic sets. These
FUPs and their associated conjectures are detailed by Dyatlov [Dya19].
In addition to being of theoretical interest, FUPs for ellipsephic sets
have been applied to derive spectral gaps for open quantum baker’s maps
[DJ18]. However, most of the work related to FUPs as yet has dealt with
the theory and applications of continuous FUPs. It is our hope that this
thesis has shed light on the discrete setting and will spur further research
in the same.
37
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39
A Numerical experiments
In this appendix, we provide some details concerning the numerical exper-
iments conducted in our work.
The Julia programming language was used to perform all the computa-
tions and to produce their associated �gures.
The FUP exponents plotted in Figures 2.1 and 2.2 were approximated
by computing βk(M,A) for k � blogM 5000c so that k ≥ 3 for the bases
considered. This experiment was a reproduction of one performed by
Dyatlov and Jin [DJ17].
In Figure 3.1, we took k � max{blogM 5000c , 3} and plotted βk for all
A ≥ 2 such that M � 4bA1/δc ≤ 75.
In Figure 3.2, we plotted β1 for the sequence of As given by A1 � 2 and
A j+1 � d1.25A je so that the resulting data points would be approximately
evenly spaced horizontally. We also imposed the limits M ≤ 105
and
β1 ≤ 10−13
(the latter because rounding errors began to dominate at that
level).
In Figure 3.3, the ellipsephic compression being approximated has
order k � 4 and its underlying ellipsephic set is of the same type as those
considered in Figures 3.1 and 3.2. Namely, A � 4, M � 4bA1/δc, and δ �7
8.
The veri�cation alluded to in Section 4.2 was essentially a brute-force
search executed on Compute Canada’s distributed computing clusters.
We did not attempt to extend the range of M any further because of
the unlikeliness of a counterexample existing for small M and our own
expectation that none exist.
40
