A Survey of Complex Dimensions, Measurability, and - - PowerPoint PPT Presentation

A Survey of Complex Dimensions, Measurability, and Lattice/Nonlattice Dichotomies

John A. Rock

Cal Poly Pomona jarock@cpp.edu Featuring various collaborative efforts of:

• M. L. Lapidus and M. van Frankenhuijsen

as well as

• H. Maier, D. ˇ

Zubrini´ c, R. de Santiago, S. Roby,

• R. Morales, C. Sargent, K. Dettmers, and R. Giza

3rd Bremen Winter School and Symposium 31 March 2015

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 1 / 79

Sound check

σ it 1 1/2 log 2 3 1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

“Can one hear the shape of an ordinary fractal string?” [LaMa] {s ∈ C : ζ(s) = 0, 0 < Re(s) < 1} ⊂ {s ∈ C : Re(s) = 1/2}

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 2 / 79

The Riemann zeta function

The Riemann zeta function is given by ζ(s) =

∞

• n=1

1 ns = 1 + 1 2s + 1 3s + 1 4s + 1 5s + 1 6s + 1 7s + 1 8s + 1 9s + · · · where s ∈ C. Naturally defined for Re(s) > 1, the Riemann zeta function has a meromorphic extension to C.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 3 / 79

The Riemann Hypothesis

σ it 1 1/2 log 2 3 1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

The nontrivial zeros of the Riemann zeta function have real part 1/2. {s ∈ C : ζ(s) = 0, 0 < Re(s) < 1} ⊂ {s ∈ C : Re(s) = 1/2}

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 4 / 79

A little history

1859 – Riemann first mentions the conjecture in his paper On the Number of Primes Less Than a Given Magnitude. 1901 – von Koch proves that the Riemann Hypothesis is equivalent to a stronger version of the prime number theorem. 1995 – Lapidus and Maier [LaMa] provide a restatement of the Riemann hypothesis in terms of the geometric and spectral

• scillations of fractal strings, stemming from the question “Can one

hear if a fractal string is Minkowski measurable?” 1997 – Lapidus and van Frankenhuysen [LvF] show that the Riemann zeta function does not have an infinite sequence of critical zeros in arithmetic progession. 2011 – Lapidus, Pearse, and Winter [LPW] find pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 5 / 79

Ordinary fractal strings

. . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

Figure : Construction of the Cantor set C and the Cantor string [0, 1] \ C.

Definition (LvF)

An ordinary fractal string Ω is a bounded open subset of the real line.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 6 / 79

Minkowski dimension

Definition

Let r ≥ 0. The upper and lower r-dimensional Minkowski contents of a bounded set A ⊂ Rm are given, respectively, by M r∗(A) = lim sup

ε→0+

volm(Aε) εm−r , M r

∗ (A) = lim inf ε→0+

volm(Aε) εm−r . (1) The upper and lower Minkowski dimensions of a bounded set A are defined by dimMA = inf{r ≥ 0 : M r∗(A) = 0} = sup{r ≥ 0 : M r∗(A) = ∞}, (2) dimMA = inf{r ≥ 0 : M r

∗ (A) = 0} = sup{r ≥ 0 : M r ∗ (A) = ∞}.

(3) If dimMA = dimMA, the common value is called the Minkowski dimension of A and is denoted dimM A.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 7 / 79

Minkowski content and measurability

Definition

Let A ⊂ Rm be such that DM = dimM A exists. The upper and lower Minkowski contents of A are denoted as M ∗(A) = M ∗DM (A), and (4) M∗(A) = M DM

∗

(A). (5) If 0 < M ∗(A) = M∗(A) < ∞, then A is said to be Minkowski measurable and the Minkowski content of A is given by M (A) = lim

ε→0+

volm(Aε) εm−DM . (6)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 8 / 79

Minkowski dimension for ordinary fractal strings

Theorem (BesTa)

If an ordinary fractal string Ω ⊂ [a, b] is of total length b − a and comprises an infinite number of open intervals (whose lengths define a nonincreasing sequence L = (ℓj)∞

j=1), then

dimM(∂Ω) = DL := inf   t ∈ R :

∞

• j=1

ℓt

j < ∞

   , (7) where ∂Ω = [a, b] \ Ω and dimM is (inner) Minkowski dimension.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 9 / 79

Fractal strings, zeta functions, complex dimensions

Definition (LvF)

A fractal string L is a sequence of positive real numbers such that L = (ℓj)∞

j=1

“=” {ln : ln distinct with multiplicity mn, n ∈ N} (8) where 0 < ℓj+1 ≤ ℓj ∀j and ℓj → 0. The dimension DL, geometric zeta function ζL, and complex dimensions DL of L, are given by DL := inf

• t ∈ R :
• ℓt

j < ∞

• ,

(9) ζL(s) :=

• ℓs

j =

• mnls

n,

(10) DL(W) := {ω ∈ W ⊂ C : ζL has a pole at ω}, (11) where Re(s) > DL and W is a suitable open region. If W = C, we write DL for DL(W).

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 10 / 79

The Cantor string

σ it 1 1/2 log 2 3 1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

LCS = {3−n : multiplicity 2n−1, n ∈ N}, DLCS = log 2 log 3, (12) ζLCS =

∞

• n=1

2n−13−ns = 3−s 1 − 2 · 3−s , and (13) DLCS = {DLCS + pki : k ∈ Z}, where p = 2π log 3. (14)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 11 / 79

Geometric counting functions

Figure : Plot of NLCS(x)/xDCS. (Think of Minkowski content as x → ∞.)

Definition (LvF)

For x > 0, the geometric counting function of a fractal string L is given by NL(x) := #{j ∈ N : ℓ−1

j

≤ x} =

• n∈N, l−1

n ≤ x

mn. (15)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 12 / 79

Zeta functions and counting functions

Figure : It is well known that the Cantor set C is not Minkowski measurable.

Theorem (LvF)

Let L be a fractal string with geometric counting function NL. Then for Re(s) > DL we have ζL(s) =

• mnls

n = s

∞ NL(x)x−s−1dx. (16)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 13 / 79

Counting functions over complex dimensions

Theorem (LvF)

Let L be a fractal string such that DL(W) consists entirely of simple

• poles. Then, under certain growth conditions (see the next two slides) on

ζL, we have NL(x) =

• ω∈DL(W)

xω ω res(ζL(s); ω) + {ζL(0)} + R(x), (17) where R(x) is an error term of small order and the term in braces is included only if 0 ∈ W\DL. For the Cantor string LCS we have NLCS(x) = 2n − 1 = 1 2 log 3

• z∈Z

xD+izp D + izp − 1 (18) where D = log3 2, p =

2π log 3, and n = [log3 x] (integer part of log3 x).

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 14 / 79

Languid fractal strings

Definition (LvF)

A fractal string L is said to be languid if ζL satisfies the following conditions: There exist constants κ, C > 0 and a two-sided sequence {Tz}z∈Z of real numbers where T−n < 0 < Tn for n ≥ 1, and lim

n→∞ Tn = ∞,

lim

n→∞ T−n = −∞,

lim

n→+∞

Tn |T−n| = 1 (19) such that for a suitable W and ‘screen’ S = ∂W we have L1 for all n ∈ Z and all σ ≥ S(Tn), |ζL(σ + iTn)| ≤ C · (|Tn| + 1)κ, (20) L2 for all t ∈ R, |t| ≥ 1, |ζL(S(t) + it)| ≤ C · |t|κ. (21)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 15 / 79

Strongly languid

Definition (LvF)

A fractal string L is said to be strongly languid if, in addition to L1 (for every σ ∈ R in (20)), ζL satisfies the following condition: There exists a sequence of screens Sm : t → Sm(t) + it for m ≥ 1, t ∈ R, with sup Sm → −∞ as m → ∞ and with a uniform Lipschitz bound supm≥1SmLip < ∞, such that L2’ There exist constants A, C > 0 such that for all t ∈ R and m ≥ 1, |ζL(Sm(t) + it)| ≤ CA|Sm(t)|(|t| + 1)κ. (22)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 16 / 79

Minkowski measurability and complex dimensions

Theorem (LvF)

Let Ω be an ordinary fractal string that is languid for a screen passing between the vertical line Re(s) = D and all of the complex dimensions with real part strictly less than D, and not passing through 0. Then the following are equivalent:

1 D is the only complex dimension with real part D = DL, and it is

simple.

2 NL(x) = cxDL + o(xDL) as x → ∞, for some positive constant c. 3 ∂Ω, the boundary of Ω, is Minkowski measurable. John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 17 / 79

The Cantor set is NOT Minkowski measurable

σ it 1 1/2 log 2 3 1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 18 / 79

Meromorphic extension in the self-similar case

Theorem (LvF)

Let Φ = {ϕj}N

j=1 be a self-similar system on R that satisfies the open set

condition with perfect, nowhere dense attractor A. Consider the ordinary fractal string [inf A, sup A] \ A with associated lengths denoted by L. Then ζL has a meromorphic extension to all of C given by ζL(s) = K

k=1 gs k

1 − n

j=1 rs j

(23) where rj is the scaling ratio associated with ϕj and gk are the lengths of the gaps (closed intervals) from the construction of A.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 19 / 79

Self-similar strings

Definition (LvF)

A self-similar string is an ordinary fractal string of the form [inf A, sup A] \ A where A is the attractor of a self-similar system on R that satisfies the open set condition.

Lemma (LvF)

Self-similar strings are strongly languid. This lemma follows from the fact that, in this case, ζL has a meromorphic continuation to all of C and so ζL satisfies L1 and L2’ with κ = 0 and A = L−1g−1

κ max {rj}.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 20 / 79

Moran’s Theorem

Theorem (Moran)

Let A ⊂ Rm be the attractor of a self-similar system Φ that satisfies the

• pen set condition with scaling vector r = (rj)N

j=1. Let DΦ denote the

unique real-valued solution of the corresponding Moran equation given by

N

• j=1

rs

j = 1,

s ∈ C. (24) Then dimH A = dimM A = DΦ, (25) where dimH denotes Hausdorff dimension and dimM denotes Minkowski dimension.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 21 / 79

Examples of attractors

Figure : The Cantor set C, Sierpinski gasket G, and the Barnesly fern. The latter is the attractor of an iterated function system, but it is technically not self-similar.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 22 / 79

Lattice and nonlattice

Definition

Let Φ be a self-similar system with scaling vector r = (rj)N

j=1 and attractor

• A. Then Φ, r, and A are said to be lattice if there exists an 0 < r < 1 and

integers kj such that rj = rkj for j = 1, 2, ..., N. Otherwise Φ, r, and A are said to be nonlattice. (Similar definitions apply to self-similar strings.) lattice: r1 = 1 2, 1 2, 1 8, 1 4

• =

1 21 , 1 21 , 1 23 , 1 22

• (26)

nonlattice: r2 = 1 2, 1 3, 1 8, 1 4

• =

1 21 , 1 2log2 3 , 1 23 , 1 22

• (27)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 23 / 79

Lattice/nonlattice dichotomy of self-similar strings

The following are (partial) reformulations of various theorems of [LvF] which, taken together, summarize key aspects of the lattice/nonlattice dichotomy of self-similar strings.

Theorem (LvF)

(i) [The boundary of a] lattice self-similar string is never Minkowski measurable and always has multiplicatively periodic oscillations of

• rder D, its dimension, in its geometry.

(ii) [The boundary of a] nonlattice self-similar string is Minkowski measurable. (iii) Every nonlattice self-similar string is approximately lattice. That is, every nonlattice self-similar string is quasiperiodic.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 24 / 79

The golden string

Consider the scaling ratios of the nonlattice self-similar system given by r = 1

2, 1 2φ

• , where φ = 1+

√ 5 2

(the golden ratio). Approximate φ by ratios of consecutive Fibonacci numbers: f0 = f1 = 1 and fM = fM−1 + fM−2 for M ≥ 2. It is well known that fM+1

fM

→ φ as M → ∞. Approximate r by rM = 1 2, 1 2fM+1/fM

• where the common ratio is

1 21/fM and we take k1 = fM and k2 = fM+1.

Then rM =

• rk1, rk2

=

• 1

21/fM fM ,

• 1

21/fM fM+1 = 1 2, 1 2fM+1/fM

• (28)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 25 / 79

Weights of self-similar systems

Definition

The weight vector of a self-similar system is defined as w = (w1, . . . , wN) = (− log r1, . . . , − log rN) (29) and the components wj are called weights.

Lemma

A scaling vector r is nonlattice if and only if at least one ratio of weights

• f the form wj/w1 is irrational.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 26 / 79

Simultaneous Diophantine Approximation

The following is a reformulation of the Diophantine approximation in [LvF] (which is a reformulation of Theorem 1A and the remark following Theorem 1E in [Schm]): If at least one of α1, . . . , αN is irrational, then they can be approximated by rational numbers with a common

• denominator. Thus, one can find integers q such that for each j, the

product qαj is close to the nearest integer.

Lemma

Let w1, w2, ..., wN be positive weights such that at least one wj/w1 is

• irrational. Then for every δ > 1, there exist integers q and k1, ..., kN such

that 1 ≤ q < δN−1 and |qwj − kjw1| ≤ w1/δ for j = 1, ..., N. Note that |qwj − kjw1| = 0 when wj/w1 is irrational, so q → ∞ as δ → ∞.

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Lattice approximation of scaling ratios

The proof of the following lemma (an application of the previous lemma to the context of the approximation of the components of scaling vectors) is a significantly simplified version of the proof of Theorem 3.18 of [LvF].

Lemma

Let r = (rj)N

j=1 be a nonlattice scaling vector. Then there exists a

sequence (rM)∞

M=1 of lattice scaling vectors, each with N components,

such that rM → r componentwise as M → ∞.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 28 / 79

Code

It is relatively easy to encode this simultaneous Diophantine approximation (work of R. Giza and K. Dettmers): Input v = [π, e, e3] ≈ [3.1416, 2.7183, 20.0855] Input q = 10 (tolerance taken as 1/q) Output ˜ v = 223 71 , 193 71 , 1426 71

• ≈ [3.1408, 2.7183, 20.0845]

For a common scaling ratio 0 < r < 1, rM =

• (r1/71)223, (r1/71)193, (r1/71)1426

≈

• rπ, re, re3

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Lattice approximation of roots of Dirichlet polymnomials

The following is a snippet of Theorem 3.6 in [LvF]. Here we have a Dirichlet polynomial f(s) = 1 −

N

• j=1

rs

j

(30) with D denoting the unique positive real root of f.

Theorem (LvF)

. . .In the nonlattice case, all roots of f, except D, have real part less than

• D. The complex roots of f can be approximated by the complex roots of

a sequence of lattice equations with larger and larger oscillatory period. Hence, the complex roots of a nonlattice (Moran) equation have a quasiperiodic structure. Furthermore, there exists a neighborhood W containing Re(s) ≥ D, such that 1/f satisfies L1 and L2 with κ = 0, and the complex roots of f in the corresponding W are simple.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 30 / 79

Lattice approximation of complex dimensions

Theorem (LvF)

Let Ω be a (lattice or nonlattice) self-similar string with ζL(s) = K

k=1 gs k

1 − n

j=1 rs j

, s ∈ C. (31) Under certain mild conditions, the complex dimensions DL are given by DL =

• ω ∈ C :

N

• j=1

rω

j = 1

• .

(32) Moreover, there is a sequence of lattice self-similar strings ΩM with complex dimensions DM such that DM → D as M → ∞ in the sense of the previous theorem.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 31 / 79

Lattice approximation of complex dimensions

1 50 p −1

f » 2/1

50 p −1

f » 3/2

1 50 p −1

f » 5/3

1 p −1

f » 8/5

1 p −1

f » 13/8

1 50 p −1

f » 21/13

1 100 p −1

f » 34/21

1 100 p −1

f » 55/34

1 100

D D D D D D D D 100 700 1 D

Thanks to Erin Pearse for the use of these images!

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 32 / 79

Collage Theorem

Proofs of the following can be found in [Falc].

Theorem

Let Φ be an iterated function system on Rm with attractor A and scaling vector r = (rj)N

j=1 such that rj < c for some c < 1, and let E be a

nonmepty compact set. Then dH(E, A) ≤

• 1

1 − c

• dH(E, Φ(E)).

(33)

Corollary

Let E be a nonempty compact set and ε > 0. There exists a self-similar set F such that dH(E, F) < ε.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 33 / 79

Characterization of Contracting Similarities

Theorem (Hut)

A map ϕ on Rm is a contracting similarity with scaling ratio r if and only if ϕ(·) = rQ(·) + t0 (34) where Q is a rotation matrix, t0 is a fixed translation vector, and 0 ≤ r < 1.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 34 / 79

Lattice approximation in the Hausdorff metric

The following theorem is the result of undergraduate research conducted with K. Dettmers and R. Giza.

Theorem

Let Φ = {ϕj}N

j=1 be a self-similar system on Rm with attractor A. Then

there exists a sequence of lattice self-similar systems (ΦM)∞

M=1 = ({ϕM,j}N j=1)∞ M=1 with attractors (AM)∞ M=1 such that

AM → A in the Hausdorff metric as M → ∞. In a proof of this theorem, the scaling ratios of the ΦM may be taken to be those given simultaneous Diophantine approximation above.

Corollary

Let E be a nonempty compact set and ε > 0. There exists a lattice self-similar set F such that dH(E, F) < ε. (Actually, this follows immediately from the Collage Theorem).

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 35 / 79

A small portion of the proof

For each j = 1, · · · , N we have ϕj(·) = rjQj(·) + tj. (35) For each j = 1, . . . , N and each M ∈ N, define rM,j using the simultaneous Diophantine approximation given above. Define ΦM = {ϕM,j}N

j=1 for each M ∈ N by

ϕM,j(·) = rM,jQj(·) + tj (36) for each j = 1, · · · , N. With the attractor of ΦM denoted AM, we have AM → A in the Hausdorff metric (and under certain conditions DM → D in the above sense) as M → ∞.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 36 / 79

The Fibonacci string

4 2 1 1 1/2 1/4 1/4 1/8 1/8 1/16

Figure : Construction of the Cantor-like boundary of the Fibonacci string interpreted as a lattice self-similar string.

The dimension D = DLFib = log2 φ of the Fibonacci string LFib is the unique real-valued solution of the Moran equation 2−2s + 2−s = 1 for s ∈ C. Also, the complex dimensions are given by the complex roots of this Moran equation, where p = 2π/ log 2: DLFib = {D + izp : z ∈ Z} ∪ {−D + i(z + 1/2)p : z ∈ Z}. (37)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 37 / 79

Oscillations of the Fibonacci string

Figure : Plot of NLFib(x)/xDLFib for the Cantor-like boundary of the Fibonacci string interpreted as a lattice self-similar string.

Oscillations in the geometric counting function NLFib are encoded in the poles of the geometric zeta function ζLFib given by ζLFib(s) =

∞

• n=0

Fn+12−ns = 1 1 − 2−s − 4−s , (38)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 38 / 79

Linear recurrence relations

Definition

A sequence {sn}∞

n=1 ⊂ C is said to satisfy a linear recurrence relation R if

there exists a1, . . . , ad ∈ C, ad = 0 such that sn = a1sn−1 + · · · + adsn−d ∀n ≥ d. (39) The number d is called the degree of linear recursion and the constant vector a := (a1, . . . , ad) is called the kernel of R. The polynomial charR(x) = xd − a1xd−1 − · · · − ad (40) is called the characteristic polynomial of R.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 39 / 79

Recursive structure of lattice strings

Definition (dSan,dSLRR)

Let Ω be a lattice self-similar string with lengths L such that for some 0 < r < 1 and positive integers kj the scaling ratios satisfy rj = rkj for each j = 1 . . . N. For L, such a value r is called a multiplicative generator with degree d = max{kj : rj = rkj}. The kernel a of L is defined as the vector with components given by ak = πk(a) = #{j : rj = rk, j ∈ {1, . . . , N}}. (41) where k ∈ {1, . . . , d}. The corresponding linear recurrence relation determined by a, denoted RL, is called the linear recurrence relation of L.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 40 / 79

Lattice strings and recurrence relations

Theorem (dSan,dSLRR)

Let Ω be a lattice self-similar string with lengths L, multiplicative generator r, kernel a, and gaps g1, . . . , gK. Then L =

K

• k=1

{gkrn : gkrn has multiplicity sn, n ∈ N ∪ {0}}, (42) where (sn)∞

n=0 is the sequence generated by RL with initial terms

(0, . . . , 0, 1). In this case, the corresponding Moran equation with s ∈ C is given by

d

• j=1

ajrjs = 1. (43)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 41 / 79

Complex dimensions of lattice strings

Theorem (dSan,dSLRR)

Suppose L denotes the lengths of lattice self-similar string with complex dimensions DL, multiplicative generator r, and kernel a. Then {ω ∈ DL : Re(ω) = DL} = log ψ − log r + 2πik log r : k ∈ Z

• .

(44) where ψ is the unique positive root of charRL.

Corollary

A lattice self-similar string is never Minkowski measurable.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 42 / 79

Minkowski measurability in higher dimensions

Theorem (LPW)

Let S ⊂ Rm be a self-similar set with Minkowski dimension D ∈ (m − 1, m) that satisfies the open set condition and whose tiling satisfies certain other conditions. Then S is Minkowski measurable if and

• nly if S is nonlattice.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 43 / 79

Distance and tube zeta functions

Definition (LRaZ)

Let A ⊂ Rm be bounded. The distance and tube zeta function of A, denoted ζd and ˜ ζA, are defined by ζd(s) =

• Aε

d(x, A)s−mdx, ˜ ζA(s) = ε ts−m−1|At|dt. (45) for Re(s) > Dd, where Dd = Dd(A) denotes the abscissa of convergence

• f the distance zeta function ζd (as well as ˜

ζA) and ε > 0 is fixed.

Theorem (LRaZ)

If A ⊂ Rm and Re(s) > dimBA, then for any ε > 0, ζd(s) = εs−m|Aε| + (m − s)˜ ζA(s). (46) Also, if D = dimM A exists, then D = Dd.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 44 / 79

A relation with Minkowski measurability

Theorem (LRaZ)

Let A ⊂ Rm be a bounded set such that there exist α, M > 0 and D ≥ 0 satisfying volm(At) = tm−D(M + O(tα)) as t → 0+. (47) Then dimM A exists, D = dimM A, A is Minkowski measurable with M D(A) = M , Dd = D, and there exists a meromorphic extension of ˜ ζA to (at least) {Re(s) > D − α}. Also, the only pole of ˜ ζA in {Re(s) > D − α} is s = D, which is simple, and res( ˜ ζA, D) = M .

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 45 / 79

A generalized notion of content

Definition (Mor)

Let f : (0, ∞) → [0, ∞) be nondecreasing. The exponent of f, denoted αf, is the nonnegative extended real number given by αf = inf{α ≥ 0 : f(x) = O(xα) as x → ∞}. (48) In the case where αf < ∞, the upper and lower (generalized) content of f are respectively given by C ∗ = lim sup

x→∞

f(x) xαf and C∗ = lim inf

x→∞

f(x) xαf . (49) If C ∗ = C∗, then f is said to be steady and the content of f, denoted C , is defined to be the common value C = C ∗ = C∗.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 46 / 79

More zeta functions and complex dimensions

Definition (Mor)

Suppose f : (0, ∞) → [0, ∞) and suppose there exists x0 > 0 such that f(x) = 0 for all x ∈ (0, x0). Then the dimension, zeta function, and complex dimensions of f, denoted Df, ζf, and Df, are given by Df = inf

• t ∈ R : t

∞ f(x)x−t−1dx < ∞

• ,

(50) ζf(s) = s ∞ f(x)x−s−1dx, and (51) Df(W) = {w ∈ W : ζf has a pole at w}, (52) where Re(s) > Df and W is a suitable open region. If W = C, we write Df for Df(W).

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 47 / 79

A steadiness (measurability) criterion

Theorem (Mor)

Suppose f : (0, ∞) → [0, ∞) such that αf < ∞ and f is nondecreasing. Then αf = Df.

Theorem (Mor)

Let f be such that ζf is (suitably) languid. Then the following are equivalent: (i) Df is the only complex dimension with real part Df, and it is simple. (ii) f(x) = E · xDf + o(xDf ) for some positive constant E. (iii) f is steady.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 48 / 79

Another example

1x1x1 4x1/4x1/4 16x1/16x1/16

The first three steps in the construction of the attractor F of Φ = {Φj}4

j=1 (see next slide).

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 49 / 79

A pair of examples

Consider the attractor F of Φ = {Φj}4

j=1 on [0, 1]2 ⊂ R2 given by:

Φ1(v) = 1 4v, Φ2(v) = 1 4v + 3 4, 0

• ,

Φ3(v) = 1 4v + 3 4, 3 4

• , and

(53) Φ4(v) = 1 4v +

• 0, 3

4

• .

(54) Consider the unit interval I = [0, 1] × {0} as the attractor of Ψ = {Ψj}4

j=1 on [0, 1]2 ⊂ R2 given by:

Ψ1(v) = 1 4v, Ψ2(v) = 1 4v + 1 4, 0

• ,

Ψ3(v) = 1 4v + 1 2, 0

• , and

(55) Ψ4(v) = 1 4v + 3 4, 0

• .

(56)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 50 / 79

Counting boxes

8 m = m = m = m = 1 2 3 4 2 1 4 1 1 2 3 4

An example of NB(A, x).

Definition

Let A ⊂ Rm and x > 0. The box-counting function of A, NB(A, x), is the maximum number of disjoint closed balls B(a, x−1) with centers a ∈ A of radius x−1. Consider the range of NB(A, ·) to be a strictly increasing sequence of positive integers denoted (Mn)n∈N.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 51 / 79

NB(F, x)

1x1x1 4x1/4x1/4 16x1/16x1/16

For the self-similar set F, we have M1 = 1 (which is always the case for any bounded set A ⊂ Rm) and M2 = 2.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 52 / 79

M3 = NB(F, 8/ √ 17) = 3

M3 = NB(F, 8/ √ 17) = 3

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M4 = NB(F, 2) = 4

M4 = NB(F, 2) = 4

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 54 / 79

M5 = NB(F, 8/ √ 2) = 8 = 4M2

M5 = NB(F, 8/ √ 2) = 8 = 4M2

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 55 / 79

Box-counting zeta functions and complex dimensions

Definition (LRoZ)

For A and n ∈ N, let m1 := M2, mn := Mn+1 − Mn (n ≥ 2), and ln := (sup{x ∈ (0, ∞) : NB(A, x) = Mn})−1. (57) The box-counting fractal string LB of A is given by LB := {ln : ln has multiplicity mn, n ∈ N}. (58) The box-counting zeta function and complex dimensions of of A, denoted ζB and DB, respectively, are the geometric zeta function of the L = LB: ζB := ζLB, DB := DLB, and DB := DLB. (59)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 56 / 79

Results from box-counting fractal strings

Proposition (LRoZ)

Let A be an infinite subset of Rm with box-counting fractal string LB and box-counting function NB(A, x). Then for x ∈ (l−1

1 , ∞) \ (l−1 n )n∈N,

NLB(x) = NB(A, x). (60)

Theorem (LRoZ)

Let A be a bounded infinite subset of Rm. Then dimBA = DB.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 57 / 79

δ-disjoint and error term

1x1x1 4x1/4x1/4 16x1/16x1/16

Let Φ = {ϕj}N

j=1 be a self-similar system with attractor S and scaling

ratios r = (rj)N

j=1. (Terminology from [Sar], concepts from [Lal].)

Definition (Sar)

A self-similar set S ⊂ Rm is δ-disjoint if the ϕj(S) are pairwise disjoint and δ := sup{α : d(x, y) > α, x ∈ ϕj(S), y ∈ ϕk(S), j = k} (61) is finite and positive.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 58 / 79

Box-counting functions, δ-disjoint

Lemma (Lal)

Let S ⊂ Rm be a δ-disjoint self similar set. Then for any x > 0, NB(S, x) =

N

• j=1

NB(S, rjx) + L(x) (62) where L(x) is an integer valued step function with finite range bounded below by 1 − N that vanishes for x > δ−1.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 59 / 79

Box-counting zeta functions, δ-disjoint

Theorem (Sar)

Suppose Φ is a δ-disjoint self-similar system with attractor A ⊂ Rm and scaling ratios rj, n = 1, . . . , N. Let LB be the box-counting fractal string

• f A with first length ℓ1. Then the box-counting zeta function of S is

given by ζB(s) = h(s) 1 − N

j=1 rs j

(63) where h(s) := ℓs

1

 

N

• j=1

(1 − rs

j)

  + s ∞

ℓ−1

1

L(x)x−s−1dx (64) is an entire function.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 60 / 79

ζB and DB of F

Example (LRoZ)

For F as above we have ζB(s) = √ 2 2 s + √ 2/2 s + √ 17/8 s + (1/2)s 1 − 4 · 4−s . (65) and DB =

• 1 + 2πiz

log 4 : z ∈ Z

• .

(66) F is not Minkowski measurable and it has nonreal box-counting complex dimensions ω with real part Reω = 1.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 61 / 79

The Riemann zeta function?

Example (LRoZ)

For the unit interval I = [0, 1] × {0}, we have NB(I, x) = ⌈x/2⌉ (the ceiling function of x/2). Hence, the box-counting zeta function of I is ζB(s) = 1 2s +

∞

• n=1

1 (2n)s = 1 2s + 1 2s ζ(s), (67) where ζ(s) is the Riemann zeta function. It is well known that ζ has a pole at s = 1 and its meromorphic extension to C does not have any other pole with real part 1. Also, I = [0, 1] is Minkowski measurable.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 62 / 79

I = [0, 1] is measurable

Figure : Plot of NB(x)/x for the unit interval I = [0, 1].

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 63 / 79

A box-counting lattice/nonlattice dichotomy

Theorem (Lal)

Let Φ be a self-similar system that satisfies the strong open set condition with attractor S where dimM S = D. (a) [Nonlattice] If the additive group generated by {log rj} is dense in R, then there is C > 0 such that NB(S, x) ∼ CxD as x → ∞. (68) (b) [Lattice] If the additive group generated by {log rj} is hZ (h > 0), then for each β ∈ [0, h) there is Cβ > 0 such that NB(S, enh−β) ∼ CβeD(nh−β) as n → ∞. (69)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 64 / 79

Complex dimensions (CDs) and measurability

Remark

Nonlattice ↔ measurable ↔ no nonreal CDs of order D Lattice and Cβ1 = Cβ2 ↔ nonmeasurable ↔ ∃ nonreal CDs of order D

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 65 / 79

The Sierpinski gasket

Example (Sar)

The box-counting function of the Sierpinkski gasket G is given by NB(G, x) =      1, 0 < x ≤ 2, 3, 2 < x ≤ 4,

3n+3 2

, 2n < x ≤ 2n+1 for n ≥ 2. (70)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 66 / 79

The Sierpinski gasket, continued

Example

The Sierpinski gasket G is not δ-disjoint for any δ > 0 and L(x) = −3 for all large enough x. Moreover, DB = {0} ∪

• log2 3 + 2πiz

log 2 : z ∈ Z

• .

(71) Also, G is Minkowski nonmeasurable (as per [LPW] for instance) and has nonreal box-counting complex dimensions where Reω = log2 3.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 67 / 79

Box-counting dimension

Definition

Let r ≥ 0. The upper and lower r-dimensional box-counting contents of a bounded set A ⊂ Rm are given, respectively, by Br∗(A) = lim sup

x→∞

NB(A, x) xr , Br

∗(A) = lim inf x→∞

NB(A, x) xr . (72) The lower and upper box-counting dimensions of A, denoted dimBA and dimBA, respectively, are given by dimBA = lim inf

x→∞

log NB(A, x) log x , (73) dimBA = lim sup

x→∞

log NB(A, x) log x . (74) When 0 < dimBA = dimBA < ∞, the limit exists and is called the box-counting dimension of A and denoted dimB A.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 68 / 79

Box-counting content and measurability

Definition

Let A ⊂ Rm be such that DB = dimB A exists. The upper and lower box-counting contents of A are denoted by B∗(A) = B∗DB(A), and (75) B∗(A) = BDB

∗

(A). (76) If 0 < B∗(A) = B∗(A) < ∞, then A is said to be box-counting measurable and the box-counting content of A is given by the common value B∗(A) = B∗(A) B(A) = lim

x→∞

NB(A, x) xDB . (77)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 69 / 79

More on a box-counting lattice/nonlattice dichotomy

Theorem (Sar)

Let S be the attractor of a δ-disjoint self-similar system Φ. (i) If Φ is lattice, then S is not box-counting measurable and always has multiplicatively periodic oscillations of order DB in its geometry. (ii) If Φ is lattice, then S is box-counting measurable. (iii) Every nonlattice self-similar set of this type is approximately lattice in the scaling ratios and in the Hausdorff metric. A conjecture (nearly proven): Every nonlattice self-similar set of this type has complex dimensions which are approximated by those of a sequence of lattice self-similar sets.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 70 / 79

Closing remarks (following [LRoZ])

A conjecture: Let A be a bounded subset of Rm. Then A is Minkowski measurable if and only if A is box-counting measurable. Also, there exist constants 0 < c1 < c2 such that c1B(A) < M (A) < c2B(A). (78) A question: What are the “tube formulas” in the context of box-counting content? How do they relate to the usual tube formulas?

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 71 / 79

. . .

. . . if there’s time. . .

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 72 / 79

Frequencies of an ordinary fractal string

1/3 . . . 2 4 8 x x x 1/9 1/27 1/81

Given an ordinary fractal string Ω with lengths L, we can listen to its

• sound. Consider the Dirichlet Laplacian △ = −d2/dx2. An eigenvalue λ
• f △ corresponds to a normalized frequency f =

√ λ/π. The frequencies

• f L are given by

f = k · ℓ−1

j

(79) where k, j ∈ N. The total multiplicity of the frequency f is w(ν)

f

=

• j:f·ℓj∈N∗

1 =

• l:f·l∈N∗

wl = w1/f + w2/f + · · · (80)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 73 / 79

Spectral zeta function and spectral counting function

Definition (LvF)

The spectral counting function of L for x > 0 is Nν(x) = #{f ≤ x} =

• f≤x

w(ν)

f .

(81)

Definition (LvF)

The spectral zeta function of L is ζν(s) =

• k,j=1

(k · ℓ−1

j )−s =

• f

w(ν)

f f−s

(82) where f runs through the distinct normalized frequencies of L.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 74 / 79

Connection between the zeta functions

The Riemann zeta function connects the geometric and spectral zeta functions of an ordinary fractal string Ω with lengths L.

Theorem (LvF)

The spectral zeta function of L is given by ζν(s) = ζL(s)ζ(s), (83) where ζ(s) in the Riemann zeta function.

Sketch.

ζν(s) =

∞

• k,j=1

(k · ℓ−1

j )−s = ∞

• j=1

ℓs

j ∞

• k=1

k−s = ζL(s)ζ(s). (84)

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 75 / 79

Reformulation of the Riemann Hypothesis [LvF]

(RH) The Riemann Hypothesis states that the nontrivial zeros of the Riemann zeta function have real part 1/2: {s ∈ C : ζ(s) = 0, 0 < Re(s) < 1} ⊂ {s ∈ C : Re(s) = 1/2}. (85) (S) Consider a given ordinary fractal string with Minkowski dimension D ∈ (0, 1). If this string has no oscillations of order D in its spectrum, does it follow that it is Minkowski measurable?

Proposition (LvF)

The Riemann Hypothesis (RH) is true if and only if the Inverse Spectral Problem (S) has a positive answer for D = 1/2.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 76 / 79

References

[BesTa] A. S. Besicovitch and S. J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure, J. London

• Math. Soc. 29 (1954), 449–459.

[dSan] R. de Santiago, The recursive structure of lattice strings, Master’s thesis, Cal Poly Pomona, 2012. [dSLRR] R. de Santiago, M. L. Lapidus, S. A. Roby, and J. A. Rock, Multifractal analysis via scaling zeta functions and recursive structure

• f lattice strings, in: Fractal Geometry and Dynamical Systems in

Pure and Applied Mathematics, Part 1, Contemporary Mathematics,

• Amer. Math. Soc., Providence, RI, 2013.

[Falc] K. Falconer, Fractal Geometry – Mathematical foundations and applications, 2nd ed., John Wiley, Chichester, 2003. [Hut] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ.

• Math. J. 30 (1981), 713–747.

[Lal] S. P. Lalley, Packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699–709.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 77 / 79

References, continued

[LaMa] M. L. Lapidus and H. Maier, The Riemann Hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. (2) 52 (1995), 15–34. [LPW] M. L. Lapidus, E. P. J. Pearse, and S. Winter, Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. Math. 227 (2011), 1349–1398. [LRaZ] M. L. Lapidus, G. Radunovi´ c, and D. ˇ Zubrini´ c, Fractal zeta functions and complex dimensions of relative fractal drums, J. Fixed Point Theory and Applications 15 (2014), 321–378. [LRoZ] M. L. Lapidus, J. A. Rock, and D. ˇ Zubrini´ c, Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics, Part 1, Contemporary Mathematics,

• Amer. Math. Soc., Providence, RI, 2013.

John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 78 / 79

References, continued

[LvF] M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings, 2nd rev. and enl. edition (of the 2006 edition), Springer Monographs in Mathematics, Springer, New York, 2012. [Mor] R. Morales, Complex dimensions and measurability, Master’s thesis, Cal Poly Pomona, 2014. [Moran] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Math. Proc. Cambridge Philos. Soc. 42 (1946), 15–23. [Sar] C. Sargent, Box-counting zeta functions of self-similar sets, Master’s thesis, Cal Poly Pomona, 2014. [Schm] W. M. Schmidt, Diophantine Approximation, Lecture Notes in Math., vol. 785, Springer-Verlag, New York, 1980.

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