Daubechies' Time-Frequency Localization Operator on Cantor Type Sets II

We study a version of the fractal uncertainty principle in the joint time-frequency representation. Namely, we consider Daubechies' localization operator projecting onto spherically symmetric $n$-iterate Cantor sets with an arbitrary base $M>1$ and alphabet $\mathscr{A}$. We derive an upper bound asymptote up to a multiplicative constant for the operator norm in terms of the base $M$ and alphabet size $|\mathscr{A}|$ of the Cantor set. For any fixed base and alphabet size, we show that there are Cantor sets such that the asymptote is optimal. In particular, the asymptote is precise for the mid-third Cantor set, which was studied in part I. Nonetheless, this does not extend to every Cantor set as we provide examples where the optimal asymptote is not achieved.


Introduction
There are many versions of the uncertainty principle, that all, in some form, state that a signal cannot be highly localized in time and frequency simultaneously. One recent version, described by Dyatlov in [2], referred to as the fractal uncertainty principle (FUP) (first introduced and developed in [3], [4], [5]), states that a signal cannot be concentrated near fractal sets in time and frequency. Fractal sets are here defined broadly as families of time and frequency sets T (h), Ω(h) ⊆ [0, 1], dependent on a continuous parameter 0 < h ≤ 1, that are so-called δ-regular with constant C R ≥ 1 on scales h to 1 (see Definition 2.2. in [2] for details). The FUP is then formulated for this general class when h → 0. While originally formulated in the separate time-frequency representation, we search for an analogous result in the joint representation as the uncertainty principles should be present regardless of our choice of time-frequency representation.
In the separate representation, Dyatlov considers the following compositions of projections onto such time and frequency sets, χ Ω(h) F h χ T (h) , where χ E is the characteristic function of a subset E ⊆ R, and F h is a dilated Fourier . For signals f ∈ L 2 (R), the FUP then states (see Theorem 2.12 and 2.13 in [2]) that for a fixed 0 < δ < 1 there exists some non-trivial exponent β > max{0, 1/2 − δ} such that Although no further estimates for the exponent is provided in [2], Jin and Zhang have made some progress in a recent paper [6]: For families of sets T (h), Ω(h) ⊆ [0, 1] that are δ-regular with constant C R ≥ 1 on scales h to 1, they have derived an explicit estimate for β > 0 only dependent on 0 < δ < 1 and C R .
In the original statement, we might not immediately recognize (1.1) as an uncertainty principle since the parameter h is also encoded in the Fourier transform. However, if we disentangle h from the Fourier transform, (1.1) turns out to be a statement regarding localization on the sets T (h)/ √ h in time and Ω(h)/ √ h in frequency. Depending on our choice of δ, the measures |T (h)/ √ h|, |Ω(h)/ √ h| might, in fact, tend to infinity as h → 0. One prominent subfamily of fractal sets, is the Cantor sets. This includes the infamous mid-third Cantor set, but instead of subdividing into three pieces and keeping two by each iteration, we generalize and subdivide into M > 1 pieces labeled {0, 1, . . . , M − 1} and keep a fixed alphabet A of said pieces. With regard to the Cantor set construction, we could just as well speak of iterations n → ∞ rather than an h-neighbourhood that tends to zero. Or equivalently, for such scaled n-iterate Cantor sets, the intervals {I j } j that make up n-iterate all satisfy This condition will be a point of reference in the subsequent discussion.
In the joint representation, we shall consider Daubechies' localization operator, based on the Short-Time Fourier Transform (STFT) with a Gaussian window, that projects onto spherically symmetric subsets of the time-frequency plane. The non-trivial assumption of radial symmetry is effective as the Daubechies operator has a known eigenbasis, the Hermite functions, and explicit formulas for the eigenvalues. With this powerful tool available, we estimate the operator norm when projecting onto the family of generalized spherically symmetric Cantor sets with base M and alphabet A defined in a disk of radius R > 0. Other treatments of localization on sparse sets utilizing the STFT can be found in [7] and [8], where sparsity is described in terms of "thin at infinity" and the Nyquist density, respectively.
The remainder of the paper is organized as follows: In section 2, we describe Daubechies' operator and the Cantor set construction in more detail. New results are divided into two sections, 3 and 4: In the first section, we keep the base M > 1 and alphabet A fixed throughout the iterations, while in the second, we introduce the notion of an indexed Cantor set where the base and alphabet may vary.
In section 3, we let the radius R = R(n) be dependent on the iterates n. For base M, we initially consider the general case when A special case of the above restriction, similar to condition (1.2) (and also (1.3) in [1]), is Let P n (M, A ) denote Daubechies' localization operator that projects onto the n-iterate spherically symmetric Cantor set with base M and alphabet A defined in a disk whose radius R(n) > 0 satisfies (1.4). Then for some constant B > 0, we find that for n = 0, 1, 2, . . . , where |A | < M denotes the size of the alphabet. This is similar to the asymptote for the midthird Cantor set, which was estimated in part I [1] and was shown to be precise. Compared to the general case, we always have that the asymptote is precise for the alphabet A = A = {0, 1, . . . , |A | − 1} and never precise for {M − 1, M − 2, . . . , M − |A |}. Moreover, for the alphabet A , the largest eigenvalue corresponds the Gaussian eigenfunction, independent of radius and iterate, which was not established for the mid-third Cantor set. While several properties transfer naturally from the non-indexed to the indexed case in section 4, for an arbitrary indexed base and alphabet, we do not always maintain good control over the operator norm. Therefore we present some simple conditions on the bases {M j } j and alphabets {A j } j which guarantees that the operator norm decays to zero for increasing iterates n.

2.1.
Daubechies' localization operator. In order to produce a joint time-frequency representation of a signal f ∈ L 2 (R), we consider the Short-Time Fourier Transform (STFT), dependent on a fixed, non-zero window function φ : R → C. At point (ω, t) ∈ R × R, at frequency ω and time t, the STFT of f with respect to the window φ, is then given by If we assume φ ≡ 1, we retrieve the (regular) Fourier transform of f , without any timedependence. For a joint time-frequency description, we assume φ to be non-constant. In particular, if we consider windows φ ∈ L 2 (R), with φ 2 = 1, the STFT becomes an isometry onto some subspace of L 2 (R 2 ), i.e., V φ f, V φ g L 2 (R 2 ) = f, g for any f, g ∈ L 2 (R). Thus, we obtain a weakly defined inversion formula, where the original signal f is recovered from V φ f via inner products.
Daubechies' localization operator, introduced in [9], is based on the idea of modifying the STFT of f by a multiplicative weight function F (ω, t) before recovering a time-dependent signal. The purpose of the weight function is to enhance and diminish different features of the (ω, t)-domain R 2 , e.g., by projecting onto a subset of R 2 . Characterized by our choice of window φ and weight F , we denote the localization operator by P F,φ , which can be weakly defined as Since its conception, this operator has not only been studied as an operator between L 2spaces, but also more broadly as an operator between modulation spaces, and therein questions regarding boundedness and properties of eigenfunctions and egenvalues remain relevant (see [10], [11], [12]). If we stick to the L 2 (R)-context, and assume the weight to be real-valued and integrable, the operator P F,φ : L 2 (R) → L 2 (R) becomes self-adjoint, compact. In particular, this means that the eigenfunctions of P F,φ form an eigenbasis for L 2 (R), and the operator norm P F,φ op is given by the largest eigenvalue in absolute value.
Similarly to Daubechies' classical paper [9] and what was done in part I [1], we shall focus our attention to weights that are spherically symmetric, that is, for some integrable function F : R + → R, we consider Combined with a normalized Gaussian window, the eigenfunctions of P F,φ are known, with explicit formulas for the associated eigenvalues: Theorem 2.1. (Daubechies [9]) Let the weight F and window φ be given by (2.1) and (2.2), respectively. Then the eigenvalues of the localization operator P F,φ read Interestingly, it was recently shown in [13] that any Hermite function H j as window and spherically symmetric weight yield localization operators with the same eigenbasis {H k } k . Another area that is being investigated is the aptly named inverse problem, where one instead derives properties of the weight (symbol) based on knowledge of the eigenfuntions. E.g., for Daubechies' operators with a Gaussian window that project onto a simply connected domain D ⊆ R 2 , we know by [14] that if H j is an eigenfunction for some j, then D reduces to a disk centered at the origin. More general situations are studied in [15], [16].
Proceeding with the direct problem and Daubechies' classical result, formula (2.3) represents a powerful tool for analyzing the localization operator and estimating the operator norm. In the subsequent discussion, we will only consider operators on the form as in Theorem 2.1. More precisely, we consider the case when F projects onto some spherically symmetric subset E ⊆ R 2 . That is, F (r) = χ E (r) for some subset E ⊆ R + such that E = {(ω, t) ∈ R 2 | ω 2 + t 2 ∈ E}. For the sake of simplicity, we will denote the associated localization operator by P E , whose eigenvalues are given by where π · E := {x ∈ R + | xπ −1 ∈ E}. Here it is worth noting that if we fix the measure |E|, we optimize the operator norm if E corresponds to a ball centered at the origin. More precisely, π·E r k k! e −r dr ≤ π|E| 0 e −r dr = 1 − e −π|E| for k = 0, 1, 2, . . . , (2.4) which was proved in Appendix A in [1].
Since these will appear frequently, we shall denote the above integrands by f k (r) := r k k! e −r for k = 0, 1, 2, . . .
We recognize the function f k (r) as a gamma probability distribution, which is monotonically increasing for r ∈ [0, k] and decreasing for r ≥ k. Finally, observe that these operators can also be studied from the perspective of Toeplitz operators on the Fock space. In particular, in [17] Galbis considers Toeplitz operators with radial symbols and derive non-trivial norm estimates for such operators.
for set X and scalars a, b > 0. Let |A | denote the cardinality of the alphabet A . Then the measure of the n-iterate Cantor set is given by In particular, for base M = 3 and alphabet A = {0, 2}, we recognize (2.6) as the standard mid-third n-iterate Cantor set, with measure (2/3) n R. Another noteworthy alphabet, that will appear frequently in the subsequent discussion, is which we will refer to as the canonical alphabet of size |A |. Such a redistribution of the alphabet does not alter the fractal dimension, i.e., the sets C n (R, M, A ) and C n (R, M, A ) will still share the fractal dimension ln |A | ln M . For each n-iterate based in [0, R], we define a corresponding map G n,R,M,A : R → [0, 1], known as the Cantor function, given by Since G n,R,M,A (x) = G n,1,M,A (xR −1 ), we set G n,M,A := G n,1,M,A , for simplicity. It is wellknown that for the mid-third Cantor set the associated Cantor function is subadditive (see [18]). However, with an arbitrary alphabet, subadditivity can no longer be guaranteed. Instead we present a weaker version, sufficient for our purpose, utilizing the canonical alphabet (see Appendix A for details).
For A = A , the above inequality is just standard subadditivity.
For the disk of radius R > 0, centered at the origin, we consider a spherically symmetric n-iterate, based on the n-iterate Cantor set in (2.6), as a subset of the form This means we consider weights of the form F (r) = χ Cn(R 2 ,M,A ) (r) for R > 0 and n = 0, 1, 2, . . .

Localization on Generalized Spherically Symmetric Cantor set
In this section we describe the behaviour of the operator norm, P Cn(R,M,A ) op as a function of the iterates n. The results are formulated in section 3.1, with proofs and proof strategy in the subsequent sections 3.2-3.5.
3.1. Results: Bounds for the operator norm. Below we present three theorems regarding the operator norm of P Cn(R,M,A ) . The first theorem shows that P Cn(R,M,A ) op can be bounded in terms of the "first eigenvalue" λ 0 (C n (R, M, A )), thus revealing the significance of the canonical alphabet A = {0, 1, . . . , |A | − 1}.
Theorem 3.1. The operator norm of P Cn(R,M,A ) is bounded from above by

Further, for the canonical alphabet
In the next theorem we present an upper bound estimate for the operator norm P Cn(R,M,A ) op .
Theorem 3.2. There exists a positive, finite constant B only dependent on |A | and M such that for each n = 0, 1, 2, . . .
Proofs of Theorem 3.1 and 3.2 are found in section 3.3 and 3.4, respectively.
Remark. If the alphabet, A , is equal the canonical alphabet, A , then the left-hand-side of the inequality of Theorem 3.2 can be bounded from below by a non-negative constant, thus making the asymptote precise.
If we now enforce condition (1.3) on the radius R, we obtain the following result: Theorem 3.3. Suppose the radius R depends on the iterates n so that R(n) → ∞ as n → ∞ while πR 2 (n) ≤ M n for all n = 0, 1, 2, . . . Then there exist positive, finite constants B L ≤ B U only dependent on M and |A | such that (a) for an arbitrary alphabet A we have the upper bound for the canonical alphabet A = A , we also have the lower bound (c) Conversely, at any alphabet size 0 < |A | < M, there exist alphabets A such that Recall that the quantity ln |A | ln M is the fractal dimension of the Cantor set with base M and alphabet A . It should be noted that the exponent ln |A | ln M − 1 in Theorem 3.3 (a) was already suggested in [1]. Since the associated upper bound holds for all alphabets and bases, this immediately begs the question whether the asymptote is precise regardless of alphabet and base. By Theorem 3.3 (b) and also Corollary 4.1 in [1], we conclude that the asymptote is precise for the Cantor set with canonical alphabet and for the mid-third Cantor set, respectively. However, by Theorem 3.3 (c), it becomes clear that we cannot extend this result to every alphabet. A constructive proof of Theorem 3.3 (c) is found in section 3.5.
3.2. Main Tool: Relative Areas. Similarly to section 4 in [1], our main tool is the concept of relative areas, namely The relative areas measures the local effect on the integrals that define the eigenvalues λ k (. . . ) when we increase from one iterate n to the next n + 1. These are in general easier to work with rather than the eigenvalues themselves directly. Hence, we shall attempt to derive properties of the relative areas that transfer to the global behaviour of the eigenvalues.
Initially, note that A 0,M,A (s, T ) is independent of the starting point s ≥ 0, which yields the nice recursive relation for the first eigenvalue from which the relative area A 0,M,A (·, T ) attains the simple form This relative area will play a significant role throughout the subsequent discussion. We conclude this section by showing A 0,M,A (·, T ) to be monotone, and illustrate how local effects can transfer to global behaviour.
It suffices to show that the second factor in the above expression is always positive, i.e., The latest claim is evident as By monotonicity of A 0,M,A (·, T ) and the the recursive relation (3.2), it follows that the associated eigenvalue λ 0 (. . . ) is increasing as a function of the radius, i.e., Monotonicity will also prove particularly useful both in section 3.5 and section 4.
3.3. Proof of Theorem 3.1. To begin with, we compare the relative areas with the canonical alphabet, for which we have the rather remarkable result.
Firstly, we write the difference with a common denominator, which, by Fubini's theorem, yields Inserting the definition f k (r) := r k k! e −r into the above integrand, we obtain For the canonical alphabet, we can immediately conclude with the following corollary: The largest eigenvalue of the operator P Cn(R,M,A ) is λ 0 (C n (R, M, A )), and consequently the operator norm is given by Proof. Since the relative area A 0,M,A (·, T ) is independent of the starting points s ≥ 0 and bounds all {A k,M,A (s, T )} k , it is clear that which, by the relation (3.2) and observation (2.4), eventually yields Proof. Since f k (r) is monotonically decreasing for r ≥ k, it follows that which combined with the ordering in Lemma 3.5, yields the result.
By the same argument as in Corollary 3.1, we utilize Corollary 3.2 to obtain the bound Relating the shifted iterates C n (. . . ) + s to the non-shifted iterates C n (. . . ), we present an almost analogous statement to Lemma 3.5 in [1].  Proof. For simplicity, set R := πR 2 . By definition (3.1), it is straightforward to compute the relative area A 0,M,A , which inserted into the recursive relation (3.2) yields This in return means Since the product j 1 − e −RM −j (1 − e −RM −(j−1) −1 is telescoping, only the initial denominator and final numerator remain, and identity (3.7) readily follows. For the inequality case, merely note that the negative exponential function is monotonically decreasing and with the canonical alphabet A = {0, 1, . . . , |A | − 1}, we recognize the appearance of the geometric series.
Using result (3.8) for the eigenvalue λ 0 (C n (R, M, A )), we compute the asymptotes of said eigenvalue, which, combined with Theorem 3.1, concludes the proof of Theorem 3.2.
By the fact that ψ k (0) = ψ k (1) = 0, it is clear that ψ k is bounded by the spline such that Thus, the sum in claim (i) is bounded from above and below by ± ∞ j=0 h |A | (y 1/M j ), respectively. From here the proof is essentially the same as for claim (i) in Proposition 4.1 in [1]. The same goes for claim (ii), where the only modification is that 3 j is exchanged for M j > 1.

Proof of Theorem 3.3 (c)
: Counterexample to precise asymptotic estimate. For our counterexample, we shall consider the reverse canonical alphabet, that is, where M denotes the associated base of the Cantor set. As it turns out, this Cantor set construction is merely a shifted version of the Cantor set with canonical alphabet, A .  where the inner radius R inner = R inner (n, M, |A |) ≥ 0 is given by According to condition (1.3), we consider radii R(n) that tends to infinity as n → ∞, which also means that the inner radius R inner = R inner (n) → ∞ as n → ∞. Based on this inner radius, we may, in fact, exclude certain eigenvalues from being the largest. More precisely, let ⌊·⌋ denote the floor function, rounding down to the nearest integer. Since the difference between two integrands f k+1 (r) − f k (r) ≥ 0 for r ≥ k + 1, it is clear that largest eigenvalue λ k (. . . ) must have index k ≥ ⌊πR 2 inner ⌋. Proceeding, we consider the universal upper bound provided by Lemma 3.6, namely 2 k+Cn(πR 2 ,M,A ) f k (r)dr ≥ λ k (C n (R, M, A )) for k = 0, 1, 2, . . . , (3.10) which holds for all alphabets, including A = A . We begin by computing the associated relative areas as k → ∞.
for a > k.
Remark. Comparing the above limit to the relative areas of λ 0 (C n (R, M, A )) in (3.4), we find that lim k A k,M,A (ak, T ) = A 0,M,A (·, (1 − 1 a )T ). Although the limit of Lemma 3.9 has a familiar form, it must be handled with some care as it, in fact, represents a lower bound rather than an upper one. In particular, the inequality holds for T ≤ ǫk 2 and a ∈ 1, 1 + ǫ 2 .
Proof. We start with (B) and relate this to If the above integrand is always positive or always negative, this translates directly to the sign of the difference. This is the same as asking if the function ln Q(x, y) = k ln y − y is always positive or always negative for ak ≤ x ≤ ak + t ≤ y ≤ ak + T . Since the derivative (ln q) ′ (x) ≥ 0 for x ∈ [k, (1 + ǫ)k], it follows that ln Q(x, y) ≥ 0 for k ≤ x ≤ y ≤ (1 + ǫ)k and thus part (B). For part (A), we simply consider 1 + ǫ = a, which yields (ln q) ′ (x) ≤ 0 for x ≥ ak.
Now, it is tempting to suggest a common upper bound regardless of starting point s = ak: Example 3.1. Suppose L > 1 is constant and that a ∈ [1, L] and T ≤ Lk for all iterates n. By Lemma 3.10 part (B), we have that However, this estimate for the relative area is essentially the same as for λ 0 (C n (R, M, A )), only with a scaled radius, R → 1 − (2L) −1 R, which, by Theorem 3.3 (a) and (b), does not change the asymptotes.
With this example in mind, we divide the integral ∞ k f k (r)dr into a significant and insignificant part, thus, gaining more control over the starting points s = ak. In the next lemma, we show what we mean by insignificant.
Lemma 3.11. For any fixed ǫ, δ > 0, there exists a positive integer K, so that the integral Proof. Using the lower bound version of Stirling's approximation formula for the factorial √ 2πn n+ 1 2 e −n ≤ n! for n = 1, 2, 3, . . . , we determine a simplified upper bound for the summand In the final inequality, we have used that the function g A (n) := A n n is monotonically increasing for 1 ≤ n ≤ Ae −1 .
Proposition 3.2. Suppose the radius R depends on the iterates n such that R(n) → ∞ as n → ∞, and let γ > 0 be a multiplicative constant. Then for every ǫ, δ > 0, there exists a positive integer N such that for every iterate n ≥ N, we have that Proof. We consider iterations n so that the relevant indices k ≥ γ · πR 2 (n) ≥ 1. For case (A), consider any fixed iterate n 0 so that The interval size |I| of the n 0 -iterate Cantor set C n 0 (πR 2 (n), M, A ) then satisfies By Lemma 3.10 (B) and identity lim k A k,M,A (ak, T ) = A 0,M,A (·, (1 − a −1 )T ), we obtain By the recursive relation (3.2) for λ 0 (C n (. . . )), it is clear that where we have simplified the argument ǫ(1 + ǫ) −1 R ≤ √ ǫR by monotonicity of λ 0 (C n (. . . )).
Since the radius R(n) → ∞ as n → ∞, there exists a positive integer n 1 so that the integral Combining the three inequalities (3.12)-(3.14), then yields (A) with any N ≥ max{n 0 , n 1 }.
For case (B), observe first, by the subadditivity of the Cantor function G n,M,A , that Further, by Corollary 3.2 and the recursive relation (3.2) for λ 0 (C n (. . . )), we have We now apply Lemma 3.11, whereas for any δ > 0 there exists a positive integer n 2 so that from which (B) follows with any N ≥ n 2 . For both cases, we chose N ≥ max{n 0 , n 1 , n 2 }. Now fix some ǫ, δ > 0. Since R inner (n) → ∞ as n → ∞, we can apply Proposition 3.2 (A) and (B), and conclude that for some threshold iterate N 0 To see why this inequality readily yields the desired result, we insert the estimates of Theorem 3.3 (a) and (b). In particular, there exist a constant B > 0 (independent of ǫ) and threshold iterate N 1 so that From these two estimates, and the fact that P Cn(R(n),M,A ) op = λ 0 (C n (R, M, A )), we obtain Since ǫ, δ are arbitrary, and the constant B does not depend on either, we are done.

Further Generalizations: Indexed Cantor set
So far in our discussion the base and alphabet have been fixed throughout the iterations. Suppose we now instead let these two quantities vary, to obtain an even more general Cantor set construction. In particular, if we index the bases {M j } ∞ j=1 and the alphabets {A j } ∞ j=1 according to each iteration, we obtain the discrete construction M l a j ∈ A j for j = 1, 2, . . . , n , (4.1) which in return yields the continuous version where an empty product is, by convention, defined as 1. We shall refer to the above construction, C n (R, {M j } j , {A j } j ), as an indexed Cantor set.
Similarly to (2.10), we can also define an n-iterate spherically symmetric, indexed Cantor set, C n (R, {M j } j , {A j } j ), and hence define the localization operator P Cn(R,{M j } j ,{A j } j ) . With condition (1.4) in mind, a natural restriction on the radius seems to be

4.1.
Results: Sufficient decay conditions. As should be expected, for an arbitrary indexed Cantor set, we cannot guarantee that the operator norm of the associated localization operator decays exponentially or even converges to zero for increasing iterates. Below we present two theorems that reflect this. In the first theorem we present some sufficient conditions for which the operator norm indeed decays exponentially.
Further, suppose that the radius R satisfies the condition πR 2 (n) ≤ γ · M 1 M 2 . . . M n 1/2 for some finite γ > 0. Then there exist finite constants α, β > 0 only dependent on ǫ, δ and γ such that the operator norm satisfies The second theorem shows that bounded quotients such that sup j |A j | M j < 1 is not itself a sufficient condition for the localization operator to converge to zero in the operator norm. Then there exist indexed bases {M j } j and alphabets {A j } j with

4.2.
Set-up and simple example. Based on our analysis of its simpler sister-operator P Cn(R,M,A ) , we begin by deducing some analogous results for P Cn(R,{M j } j ,{A j } j ) . In particular, identity (3.2) can be replaced by Further, as a matter of additional indexing in the proofs, it follows that an analogue of Theorem 3.1 must hold for the indexed localization operator, namely By the above inequality, it suffices to establish decay conditions for the first eigenvalue with canonical alphabets. Therefore, by the recursive relation (4.4), the relative areas {A 0,M j ,A j } j are of particular interest. Recall from result (3.4) that these relative areas only depend on the base M j and alphabet A j via the quotient Hence, if |A j | M j → 1 as j → ∞ at a sufficient rate, the above product does not converge to zero as n → ∞, and so the Cantor set itself has positive measure. That is, for such a choice of bases {M j } j and alphabets {A j } j , there exists some 0 < ρ < 1 for which |C n (R, {M j } j , {A j } j )| ≥ R · ρ > 0 for all n = 0, 1, 2, . . . Using the integral formula for the first eigenvalue as a lower bound for the operator norm, it follows that This shows that for a Cantor set of positive measure lim inf n P Cn(R,... ) op > 0 when the radius R is fixed.
If we instead let R = R(n) increase according to the iterates n, we can choose canonical alphabets A j = A j , which guarantees that the first eigenvalue λ k (R(n), . . . ) is also increasing. Hence, lim inf n P Cn(R(n),M j ,A j ) op > 0.

Proofs of Theorem 4.1 and Theorem 4.2.
Since the first factor in (4.6) tends to 1 as n → ∞, convergence of the operator norm is determined by the remaining product n j=1 (. . . ). For simpler notation we assume the constant γ = 1 in both proofs.
Proof. (Theorem 4.1) Initially, note that by the interpretation as a relative area, the function A |A j | M j must also be monotonically increasing in terms of the index |A j | M j . Since the index is bounded by ǫ, we obtain the basic inequality  For these factors the argument satisfies M [... ]/2 ≤ 1, which, by monotonicity, means where |S n | denotes the cardinality of S n . Since the relative areas A ǫ (T ) ∈ (0, 1) ∀ T > 0, the remaining factors in {0, 1, . . . , n − 1} \ S n can be bounded by 1. Hence, in order to prove exponential decay in the operator norm, it suffices to verify that |S n | ≥ a · n for some constant a > 0. The latest claim is easily verified by a closer inspection of definition (4.7), where S n can be expressed as S n = 1 + δ 2 + δ n , 1 + δ 2 + δ n + 1, . . . , n − 1 .
Proof. (Theorem 4.2) We will construct a sequence of bases {M j } j and alphabet cardinalities {|A j |} j , where the product does not converge to zero as the iterate n tends to infinity: For any fixed ǫ ∈ (0, 1), chose an initial base M > 1 and alphabet size |A | < M such that |A | M ≤ ǫ. Set Proceeding, we make the following two basic observations (ii) A θ (T ) ≥ 1 − e −θT (which is still monotonically increasing) and (iii) M j ≥ M ∀ j ∈ N. Combining these two observations with product (4.8), then yields the lower bound 1 − e −θN m , where we have defined N := M 1/2 for simplicity. It remains to show that the right-handside of the above inequality does not converge to zero for any fixed N > 1 and θ ∈ (0, 1). Exchanging the product for a sum, the statement is equivalent to showing that ∞ m=2 ln 1 − e −θN m > −∞ for any fixed N > 1 and θ ∈ (0, 1).
Utilizing the lower bound ln(1 − x) ≥ − x 1−x for 0 < x < 1, we obtain By definition, we have that the n-iterate Cantor set C n effectively partitions [0, 1] into M n subintervals {I n k } M n k=1 , which we will refer to as the n-blocks. These blocks will either belong to C n or have zero intersection, i.e., |I n k ∩ C n | = |I n k |(= M −n ) or = 0. By the recursive construction, it follows that I n−1 k ∩ C n either contains an entire alphabet of n-blocks or no n-blocks at all, i.e., |I n−1 k ∩ C n | = |A |M −n or = 0. In general,  for integers 0 ≤ m j < M and parameter 0 ≤ a ≤ M 1−n , we are now able to derive a more explicit formula for G n,M,A . In particular, by recursive use of (i) and finally inserting result (v) An interval of size |I| ≤ M j−n for j ∈ N contains at most |A | j n-blocks, i.e., G n,M,A (y) − G n,M,A (x) ≤ |A | j−n .
In the case (iv), observe that I will at most intersect two (n − 1)-blocks, say I k and I k+1 .
If both intersections of I k and I k+1 with C n is non-zero, then the alphabet of n-blocks will have the same distribution in each (n − 1)-block. Hence, by (i), |I ∩ C n | = |I ∩ I k ∩ C n | + |(I − M 1−n ) ∩ I k ∩ C n | ≤ |I k ∩ C n | = |A |M −n , which yields (iv) after normalization. Result