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Minkowski dimension and nonmeasurability of lattice-type fractals

Minkowski dimension and nonmeasurability of lattice-type fractals

Erin P . J. Pearse

epearse@calpoly.edu

California Polytechnic State University

Featuring joint work with Michel Lapidus, Sabrina Kombrink, and Steffen Winter

• Mar. 6, 2015

California Polytechnic State University San Luis Obispo, CA

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Eigenvalues of the Dirichlet Laplacian

The Dirichlet problem for a bounded domain Ω ⊆ Rd:

• ∆u = λu,
• n Ω,

u = 0,

• n ∂Ω,

where ∆ = −

• ∂2

∂x2

1 + · · · + ∂2

∂x2

d

• is the Laplace operator.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Eigenvalues of the Dirichlet Laplacian

The Dirichlet problem for a bounded domain Ω ⊆ Rd:

• ∆u = λu,
• n Ω,

u = 0,

• n ∂Ω,

where ∆ = −

• ∂2

∂x2

1 + · · · + ∂2

∂x2

d

• is the Laplace operator.

The spectrum of this operator is 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . .

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Eigenvalues of the Dirichlet Laplacian

The Dirichlet problem for a bounded domain Ω ⊆ Rd:

• ∆u = λu,
• n Ω,

u = 0,

• n ∂Ω,

where ∆ = −

• ∂2

∂x2

1 + · · · + ∂2

∂x2

d

• is the Laplace operator.

The spectrum of this operator is 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . The eigenvalue counting function is N(λ) = max{j .

. . λj ≤ λ}.

l1 l2 l3=l4 l5 l6 l N(l)

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A. Theorem (Weyl, 1911). The eigenvalue counting function N(λ) = max{j .

. . λj ≤ λ} satisfies

lim

λ→∞

N(λ) λd/2 = cd vold(Ω), cd = (2π)−dωd.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A. Theorem (Weyl, 1911). The eigenvalue counting function N(λ) = max{j .

. . λj ≤ λ} satisfies

lim

λ→∞

N(λ) λd/2 = cd vold(Ω), cd = (2π)−dωd.

“My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.” — Hermann Weyl

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A. Theorem (Weyl, 1911). The eigenvalue counting function N(λ) = max{j .

. . λj ≤ λ} satisfies

lim

λ→∞

N(λ) λd/2 = cd vold(Ω), cd = (2π)−dωd.

“My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.” — Hermann Weyl “In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.” — Hermann Weyl

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A. Theorem (Weyl, 1911). Weyl’s asymptotic law: N(λ) = cdλd/2 vold(Ω) + o(λd/2), as λ → ∞.

“My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.” — Hermann Weyl “In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.” — Hermann Weyl

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry Weyl asymptotics

Weyl’s spectral asymptotic formula

Conjecture (Lorentz, 1910). One can hear the area of a drum: lim

λ→∞

2πN(λ) λ = A. Theorem (Weyl, 1911). Weyl’s asymptotic law: N(λ) = cdλd/2 vold(Ω) + o(λd/2), as λ → ∞. Conjecture (Weyl, 1911). N(λ) = cdλd/2 vold(Ω) + cd−1λ(d−1)/2 vold-1(∂Ω) + o(λ(d−1)/2), as λ → ∞.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl Conjecture

Conjecture proved by Courant (1922) with bound O(λ(d−1)/2 log λ). Levitan (1952): O(λ(d−1)/2) if Ω is compact and closed. Duistermaat and Guillemin (1975): o(λ(d−1)/2) for nice Ω. M´ etivier (1977): extended to more general elliptic operators. Ivrii (1980): generalized to milder restrictions on Ω.

(set of periodic billiards has measure 0).

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl Conjecture

Conjecture proved by Courant (1922) with bound O(λ(d−1)/2 log λ). Levitan (1952): O(λ(d−1)/2) if Ω is compact and closed. Duistermaat and Guillemin (1975): o(λ(d−1)/2) for nice Ω. M´ etivier (1977): extended to more general elliptic operators. Ivrii (1980): generalized to milder restrictions on Ω.

(set of periodic billiards has measure 0).

Mark Kac (1966): “Can one hear the shape of a drum?”

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl Conjecture

Conjecture proved by Courant (1922) with bound O(λ(d−1)/2 log λ). Levitan (1952): O(λ(d−1)/2) if Ω is compact and closed. Duistermaat and Guillemin (1975): o(λ(d−1)/2) for nice Ω. M´ etivier (1977): extended to more general elliptic operators. Ivrii (1980): generalized to milder restrictions on Ω.

(set of periodic billiards has measure 0).

Mark Kac (1966): “Can one hear the shape of a drum?” Gordon, Webb, Wolport (1992): “No.”

1 1 2 2 3 3 1 1 2 2 3 3

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl-Berry Conjecture

Michael Berry (1979): For non-smooth boundaries, change N(λ) = cdλd/2 vold(Ω) + cd−1λ(d−1)/2 vold-1(∂Ω) + o(λ(d−1)/2) to N(λ) = cdλd/2 vold(Ω) + cDλD/2HD(∂Ω) + o(λ(D−1)/2) where D is the Hausdorff dimension of the boundary.

Here, D is Hausdorff dimension (and need not be an integer). HD is the measure associated to dimension D.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl-Berry Conjecture

Michael Berry (1979): For non-smooth boundaries, change N(λ) = cdλd/2 vold(Ω) + cd−1λ(d−1)/2 vold-1(∂Ω) + o(λ(d−1)/2) to N(λ) = cdλd/2 vold(Ω) + cDλD/2HD(∂Ω) + o(λ(D−1)/2) where D is the Hausdorff dimension of the boundary. Brossard and Carmona (1986): “No. But maybe Minkowski dimension instead?”

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl-Berry Conjecture

Michael Berry (1979): For non-smooth boundaries, change N(λ) = cdλd/2 vold(Ω) + cd−1λ(d−1)/2 vold-1(∂Ω) + o(λ(d−1)/2) to N(λ) = cdλd/2 vold(Ω) + cDλD/2HD(∂Ω) + o(λ(D−1)/2) where D is the Hausdorff dimension of the boundary. Brossard and Carmona (1986): “No. But maybe Minkowski dimension instead?” Lapidus and Pomerance (1993): “Yes, for subsets of R.” N(λ) = λ vol1(Ω) + cDMD(∂Ω)λD + o(λD), as λ → ∞, where D is Minkowski dimension and MD is Minkowski content.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

The Weyl-Berry Conjecture

Michael Berry (1979): For non-smooth boundaries, change N(λ) = cdλd/2 vold(Ω) + cd−1λ(d−1)/2 vold-1(∂Ω) + o(λ(d−1)/2) to N(λ) = cdλd/2 vold(Ω) + cDλD/2HD(∂Ω) + o(λ(D−1)/2) where D is the Hausdorff dimension of the boundary. Brossard and Carmona (1986): “No. But maybe Minkowski dimension instead?” Lapidus and Pomerance (1993): “Yes, for subsets of R.” N(λ) = λ vol1(Ω) + cDMD(∂Ω)λD + o(λD), as λ → ∞, where D is Minkowski dimension and MD is Minkowski content. Kigami and Lapidus (1993): similar results for ∆ on fractals.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals. Can one hear D (the dimension of ∂Ω)? Lapidus and Pomerance (1993): “Yes.”

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let I ⊆ R be a compact interval. Let Ω ⊆ I be an open set consisting of infinitely many open intervals. Can one hear D (the dimension of ∂Ω)? Lapidus and Pomerance (1993): “Yes.” Can one hear if ∂Ω is Minkowski measurable? Lapidus and Maier (1995): “Iff ζ(s) has no zero on Re s = D.”

Here ζ(s) =

∞

• n=1

n−s is the Riemann zeta function.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let A, B ⊆ Rd be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B.

B A W

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let A, B ⊆ Rd be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B. Can one hear D (the dimension of ∂Ω)? Lapidus and van Frankenhuijsen (2000): “Yes.” Can one hear if ∂Ω is Minkowski measurable? Now measurability can fail for more than one reason. Arithmetic properties of scaling ratios. Minkowski measurability of ∂B.

Minkowski dimension and nonmeasurability of lattice-type fractals Motivation: inverse problems in spectral geometry The Weyl conjectures

Can you hear the shape of a fractal drum?

Let A, B ⊆ Rd be a bounded open set. Let Ω ⊆ A consist of infinitely many images of B. Can one hear D (the dimension of ∂Ω)? Lapidus and van Frankenhuijsen (2000): “Yes.” Can one hear if ∂Ω is Minkowski measurable? Now measurability can fail for more than one reason. Arithmetic properties of scaling ratios. Minkowski measurability of ∂B.

So there is some interest in having a robust collection of examples

• f sets which are (or are not) Minkowski measurable.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Minkowski dimension is box-counting dimension

ε = 150km ε = 75km ε = 37.5km N(ε) = 22 N(ε) = 53 N(ε) = 130

N(ε) = #{boxes of size ε required to cover the set (coastline)}.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Minkowski dimension is box-counting dimension

ε = 150km ε = 75km ε = 37.5km N(ε) = 22 N(ε) = 53 N(ε) = 130

N(ε) = #{boxes of size ε required to cover the set (coastline)}. Idea: if C is the measure of the set, then (number of boxes) · (box size) = N(ε) · (ε)D ≈ C

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Minkowski dimension is box-counting dimension

ε = 150km ε = 75km ε = 37.5km N(ε) = 22 N(ε) = 53 N(ε) = 130

N(ε) = #{boxes of size ε required to cover the set (coastline)}. Idea: if C is the measure of the set, then (number of boxes) · (box size) = N(ε) · (ε)D ≈ C = ⇒ N(ε) ≈ C 1

ε

D

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

How to find D: box-counting

N(ε) ≈ C( 1

ε)D

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

How to find D: box-counting

N(ε) ≈ C( 1

ε)D

= ⇒ log N(ε) ≈ log

• C( 1

ε)D

log N(ε) ≈ log C + D log( 1

ε) log N(ε) log( 1 ε ) ≈ log C log( 1 ε ) + D

D = lim

ε→0

log N(ε) log( 1

ε)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Next(ε) = min{J .

. . A ⊆ J j=1 B(xj, ε), where (xj)j∈J ⊆ Rd}

Nint(ε) = min{J .

. . A ⊆ J j=1 B(xj, ε), where (xj)j∈J ⊆ A}

Npack(ε) = max{J .

. . B(xj, ε) ∩ B(xk, ε) = ∅, j = k, where (xj)j∈J ⊆ A}

Nnet(ε) = max{J .

. . |xj − xk| ≥ ε, where (xj)j∈J ⊆ A}

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Next(ε) = min{J .

. . A ⊆ J j=1 B(xj, ε), where (xj)j∈J ⊆ Rd}

Nint(ε) = min{J .

. . A ⊆ J j=1 B(xj, ε), where (xj)j∈J ⊆ A}

Npack(ε) = max{J .

. . B(xj, ε) ∩ B(xk, ε) = ∅, j = k, where (xj)j∈J ⊆ A}

Nnet(ε) = max{J .

. . |xj − xk| ≥ ε, where (xj)j∈J ⊆ A}

Nnet(2ε) = Npack(ε) ≤ 2nNext(ε) and Next(ε) ≤ Nint(ε) ≤ Nnet(ε).

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Minkowski’s formulation: D = n − lim

ε→0

log VA(ε) log ε . Here, VA(ε) = vold(A−ε) is an inner tube formula, and A−ε is the inner ε-parallel set of A: A−ε = {x ∈ A .

. . d(x, A) < ε}.

A A

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Minkowski’s formulation: D = n − lim

ε→0

log VA(ε) log ε . Here, VA(ε) = vold(A−ε) is an inner tube formula, and A−ε is the inner ε-parallel set of A: A−ε = {x ∈ A .

. . d(x, A) < ε}.

A-e

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Minkowski’s formulation: D = n − lim

ε→0

log VA(ε) log ε . Here, VA(ε) = vold(A−ε) is an inner tube formula, and A−ε is the inner ε-parallel set of A: A−ε = {x ∈ A .

. . d(x, A) < ε}.

A-e So you can find D if you can find VA(ε)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Box-counting

Alternative formulations

Box-counting formulation: D = lim

ε→0

log N(ε) log( 1

ε) .

Minkowski’s formulation: D = n − lim

ε→0

log VA(ε) log ε . Here, VA(ε) = vold(A−ε) is an inner tube formula, and A−ε is the inner ε-parallel set of A: A−ε = {x ∈ A .

. . d(x, A) < ε}.

PROBLEM: The limits may not exist.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski measurability

Definition. The set A ⊆ Rd is Minkowski measurable in dimension α iff Mα(A) = lim

ε→0+ VA(ε)ε−(d−α)

exists with 0 < Mα(A) < ∞. In this case, Mα(A) is the α-dimensional Minkowski content of A.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski measurability

Definition. The set A ⊆ Rd is Minkowski measurable in dimension α iff Mα(A) = lim

ε→0+ VA(ε)ε−(d−α)

exists with 0 < Mα(A) < ∞. In this case, Mα(A) is the α-dimensional Minkowski content of A. One also has the average Minkowski content Mα(A) := lim

δ→0+

1 | ln δ| 1

δ

εα−dVA(ε)dε ε which always exists for self-similar sets [Gatzouras].

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski measurability

Definition. The set A ⊆ Rd is Minkowski measurable in dimension α iff Mα(A) = lim

ε→0+ VA(ε)ε−(d−α)

exists with 0 < Mα(A) < ∞. In this case, Mα(A) is the α-dimensional Minkowski content of A. Since Mα(A) =      ∞, α < D, MD(A), α = D, 0, D < α, let M(A) = MD(A).

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Measuring with respect to different dimensions

a-dimensional measure dimension a

a=D ¥

a(A)

D(A)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Hausdorff: Hα,ε(A) = inf{

• j∈J

|Uj|α .

. . A ⊆

• j∈J

Uj, |Uj| ≤ ε}

ε→0+

− − − − − → Hα(A)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Hausdorff: Hα,ε(A) = inf{

• j∈J

|Uj|α .

. . A ⊆

• j∈J

Uj, |Uj| ≤ ε}

ε→0+

− − − − − → Hα(A) Minkowski: N(ε)εα = inf{

• j∈J

|Uj|α .

. . A ⊆

• j∈J

Uj, |Uj| = ε}

ε→0+

− − − − − →? Mα(A) Hausdorff allows countable covers by arbitrarily sized sets. Minkowski uses finite covers by uniformly sized sets.

(Compare to partitions used in Lebesgue vs Riemann integration.)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Why Minkowski dimension instead of Hausdorff?

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Why Minkowski dimension instead of Hausdorff?

It is not as refined a notion as Hausdorff dimension!

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Hausdorff measure: σ-additive. Hausdorff dimension is sensitive to translations of connected components.

dim(¶W) = log32 dim(¶W) = 0

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Minkowski vs. Hausdorff

Hausdorff measure: σ-additive. Hausdorff dimension is sensitive to translations of connected components.

dim(¶W) = log32 dim(¶W) = 0 dim(¶W) = log32 dim(¶W) = log32

Minkowski content: only finitely additive. Minkowski dimension is invariant under translations of connected components. = ⇒ dimM makes more sense for drums

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

What VA(ε) looks like, for open subsets of R

• Definition. A fractal string L is the complement of a fractal subset
• f R. L is a bounded open subset of R, so

L = {Ln}∞

n=1,

where each Ln is an open interval with length denoted ℓn.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

What VA(ε) looks like, for open subsets of R

• Definition. A fractal string L is the complement of a fractal subset
• f R. L is a bounded open subset of R, so

L = {Ln}∞

n=1,

where each Ln is an open interval with length denoted ℓn. The Cantor string: CS = {1

3, 1 9, 1 9, 1 27, 1 27, 1 27, 1 27, 1 81, . . . } L1 L2 L3 L4 L5 L6 L7 L8

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

What VA(ε) looks like, for open subsets of R

• Definition. A fractal string L is the complement of a fractal subset
• f R. L is a bounded open subset of R, so

L = {Ln}∞

n=1,

where each Ln is an open interval with length denoted ℓn. The Cantor string: CS = {1

3, 1 9, 1 9, 1 27, 1 27, 1 27, 1 27, 1 81, . . . }

For fractal strings: VL(ε) =

• ℓn>2ε

2ε +

• ℓn≤2ε

ℓn.

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e) 6 18

6 18 54

VL(ε) =

• ℓn>2ε

2ε +

• ℓn≤2ε

ℓn

ε 2ε + 2ε + 2ε + l4 + l5 + l6 + l7 + l8 + l9 + ...

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e) 6 18

6 18 54

VL(ε) =

• ℓn>2ε

2ε +

• ℓn≤2ε

ℓn

ε 2ε + 2ε + 2ε + 2ε + l5 + l6 + l7 + l8 + l9 + ...

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e)

e1-D

6 18

6 18 54

M = lim

ε→0+ VL(ε)ε−(1−D)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e)

e1-D

6 18

6 18 54

M = lim

ε→0+ VL(ε)ε−(1−D)

1 1 1 6 18 54

* *

e

e1-D V

(e)

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e)

e1-D

6 18

6 18 54

M = lim

ε→0+ VL(ε)ε−(1−D) * * 1

54 1 1 1 2 6 18

e1-D V

(e)

loge

Minkowski dimension and nonmeasurability of lattice-type fractals Minkowski dimension Minkowski measurability

Using VL(ε) to see Minkowski measurability

1 1 1

V

(e)

e1-D

6 18

6 18 54

There are “geometric oscillations” in VL(ε) of order D. This means limε→0+ ε−(1−D)VL(ε) cannot exist. ε−(1−D)VL(ε) contains terms of the form cωεIm(ω)✐ = cωeIm(ω)✐ log ε.

Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets

How to compute the tube formula for a self-similar set?

Use inner tube formula for A∁ to find outer tube formula for A.

= ∪

Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets

How to compute the tube formula for a self-similar set?

Use inner tube formula for A∁ to find outer tube formula for A.

= ∪

(1) Obtain the components of A∁ from the IFS. (2) Determine compatibility conditions. (3) Compute V(ε) = vold((A∁)−ε) using complex dimensions.

Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets Self-similar sets

Self-similarity sets and IFSs

SG is invariant under Φj(x) = x

2 + qj, j = 1, 2, 3

where {q1, q2, q3} are the vertices of a triangle.

q1 F2(q2) = q2 q2 q3

Minkowski dimension and nonmeasurability of lattice-type fractals Tube formulas of self-similar sets Self-similar sets

Self-similarity sets and IFSs

SG is invariant under Φj(x) = x

2 + qj, j = 1, 2, 3

where {q1, q2, q3} are the vertices of a triangle.

q1 F2(q2) = q2 q2 q3

• Definition. A self-similar set F ⊆ C(X) is a fixed point

F = Φ(F) := J

j=1 Φj(F),

F = ∅. Each Φj should be contractive: |Φj(x) − Φj(y)| < |x − y|.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling

Initiating the tiling construction

For the construction to be possible, assume F satisfies the open set condition If O is a feasible open set for F, this means:

1 Φj(O) ∩ Φk(O) = ∅ for j = k, 2 Φj(O) ⊆ O for each j, 3 F ⊆ O,

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling

Initiating the tiling construction

For the construction to be possible, assume F satisfies the open set condition , and int F = ∅. If O is a feasible open set for F, this means:

1 Φj(O) ∩ Φk(O) = ∅ for j = k, 2 Φj(O) ⊆ O for each j, 3 F ⊆ O, and 4 O Φ(O).

For a feasible open set O, construct a tiling of K := O.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Two tiling examples: Koch curve and Sierpinski gasket

The Koch tiling and the Sierpinski gasket tiling

Φ1(z) = ξz, Φ2(z) = (1 − ξ)(z − 1) + 1, for ξ = 1

2 + ✐ 2 √ 3 ∈ C.

Φ1(x) = x

2 + p1, Φ2(x) = x 2 + p2, Φ3(x) = x 2 + p3.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Definition of the tiling

Each tile is the image of a generator

• Definition. The generators {Gq}Q

q=1 are the connected components

• f O \ Φ(K).

Some examples may have multiple generators.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling Definition of the tiling

Each tile is the image of a generator

• Definition. The generators {Gq}Q

q=1 are the connected components

• f O \ Φ(K).
• Definition. The self-similar tiling associated with Φ and O is

T = T (O) = {Φw(Gq) .

. . w ∈ W, q = 1, . . . , Q},

where W := ∞

k=0{1, . . . , N}k is all finite strings on {1, . . . , N}, and

Φw := Φw1 ◦Φw2 ◦. . .◦Φwn. Let T =

R∈T R denote the union of the tiles.

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators

Choosing a good O (or K)

Theorem [Compatibility Theorem]: Let int F = ∅ satisfy OSC with feasible set O and associated tiling T (O). Then TFAE:

1 bd T = F. 2 bd Gq ⊆ F, for all q ∈ Q. 3 Fε ∩ K = T−ε for all ε ≥ 0. 4 Fε ∩ K∁ = Kε ∩ K∁ for all ε ≥ 0.

So for a given Φ and F, check that one of 1–4 is satisfied. Then 5–6 ensure the inner/outer decomposition:

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators

Choosing a good O (or K)

Specific possibilities: (1) Choose K = [F] and O = int K. Feasible iff int Φj(K) ∩ Φk(K) = ∅ for j = k. (Tileset condition) In this case, int F = ∅ iff F is convex. (Nontriviality condition)

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators

Choosing a good O (or K)

Specific possibilities: (1) Choose K = [F] and O = int K. Feasible iff int Φj(K) ∩ Φk(K) = ∅ for j = k. (Tileset condition) In this case, int F = ∅ iff F is convex. (Nontriviality condition) (2) Let U be the unbounded component of F∁. Choose K = U∁ (the envelope of F) and O = int K. For the envelope, one always has bd K ⊆ F ⊆ K ⊆ [F], and K is convex iff K = [F].

Minkowski dimension and nonmeasurability of lattice-type fractals The canonical self-affine tiling The compatibility condition for generators

Choosing a good O (or K)

Specific possibilities: (1) Choose K = [F] and O = int K. Feasible iff int Φj(K) ∩ Φk(K) = ∅ for j = k. (Tileset condition) In this case, int F = ∅ iff F is convex. (Nontriviality condition) (2) Let U be the unbounded component of F∁. Choose K = U∁ (the envelope of F) and O = int K. For the envelope, one always has bd K ⊆ F ⊆ K ⊆ [F], and K is convex iff K = [F].

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula

Encoding the data into a formula

We have seen A decomposition of the set to be measured (via tiling). Conds on {Φ1, . . . , ΦJ} that ensure tiling is possible. Conds on O (or K = O) that allow vold(Fε) = VT (ε) + vold(Kε).

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula

Encoding the data into a formula

We have seen A decomposition of the set to be measured (via tiling). Conds on {Φ1, . . . , ΦJ} that ensure tiling is possible. Conds on O (or K = O) that allow vold(Fε) = VT (ε) + vold(Kε). Next: how to encode the resulting data from The scaling factors rw, where rw = rw1 . . . rwn is scaling ratio of Φw := Φw1 ◦. . .◦Φwn. The generator(s) G = O \ Φ(K).

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

The scaling data of the tiling

• Definition. Let rw = rw1rw2 . . . rwk be the scaling ratio of Φw.

The scaling zeta function is given by the scaling ratios of Φ via ζs(s) =

• w∈W

rs

w =

1 1 − J

j=1 rs j

, for s ∈ C. ζs records the sizes (and multiplicities) of the tiles T = {Φw(G)}.

W :=

∞

• k=0

{1, . . . , J}k is all finite seqs of indices, and Φw := Φw1◦Φw2◦. . .◦Φwk.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

The scaling data of the tiling

• Definition. Let rw = rw1rw2 . . . rwk be the scaling ratio of Φw.

The scaling zeta function is given by the scaling ratios of Φ via ζs(s) =

• w∈W

rs

w =

1 1 − J

j=1 rs j

, for s ∈ C. ζs records the sizes (and multiplicities) of the tiles T = {Φw(G)}. Poles of ζs encode scaling information: Ds = {ω .

. . ω is a pole of ζs}.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

The scaling data of the tiling

• Definition. Let rw = rw1rw2 . . . rwk be the scaling ratio of Φw.

The scaling zeta function is given by the scaling ratios of Φ via ζs(s) =

• w∈W

rs

w =

1 1 − J

j=1 rs j

, for s ∈ C. ζs records the sizes (and multiplicities) of the tiles T = {Φw(G)}. Poles of ζs encode scaling information: Ds = {ω .

. . ω is a pole of ζs}.

The complex dimensions Ds generalize the Minkowski dimension. Theorem [Lapidus et al.] D ∈ Ds and D = inf{σ ∈ R .

. . ζs(σ) < ∞}.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Consider a self-similar set with mappings Φ1, Φ2 having scaling ratios r1 = 2−1 and r2 = 2−φ. D is the set of solutions of the transcendental equation 2−s + 2−φs = 1 (s ∈ D),

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Consider a self-similar set with mappings Φ1, Φ2 having scaling ratios r1 = 2−1 and r2 = 2−φ. D is the set of solutions of the transcendental equation 2−s + 2−φs = 1 (s ∈ D), The fractal can be approximated by a sequence with scaling ratios 2 1, 3 2, 5 3, 8 5, 13 8 , 21 13, 34 21, 55 34, . . . − → φ. For any such approximation, r1 and r2 are integer powers of some common base r (F is lattice type) r1 = rk1, r2 = rk2, k1, k2 ∈ N

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

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

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Lattice vs. Nonlattice

For self-similar strings, a dichotomy exists.

The lattice case: {log r1, . . . , log rJ} are rationally dependent. #{Re ω} is finite. ∃ a row of dimns on Re s = D. Re ω = D for infinitely many ω ∈ Ds. ∂L is not Minkowski measurable. Example: Cantor String CS.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Lattice vs. Nonlattice

For self-similar strings, a dichotomy exists.

The lattice case: {log r1, . . . , log rJ} are rationally dependent. #{Re ω} is finite. ∃ a row of dimns on Re s = D. Re ω = D for infinitely many ω ∈ Ds. ∂L is not Minkowski measurable. Example: Cantor String CS. The nonlattice case: Some log rj are rationally independent. #{Re ω} is infinite. Re ω dense in parts of [σl, D]. Re ω = D = ⇒ ω = D. ∂L is Minkowski measurable. Example: The golden string GS.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Conjecture: Lattice/Nonlattice Dichotomy

• Theorem. A self-similar subset F ⊆ R is Minkowski measurable

⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.)

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Conjecture: Lattice/Nonlattice Dichotomy

• Theorem. A self-similar subset F ⊆ R is Minkowski measurable

⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.) Lapidus et. al conjectured that (under suitable hypotheses):

1 Lattice-type fractals are not Minkowski measurable. 2 Nonlattice-type fractals are Minkowski measurable.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Conjecture: Lattice/Nonlattice Dichotomy

• Theorem. A self-similar subset F ⊆ R is Minkowski measurable

⇐ ⇒ F is nonlattice. (Proved independently by Falconer and Lapidus.) Lapidus et. al conjectured that (under suitable hypotheses):

1 Lattice-type fractals are not Minkowski measurable. 2 Nonlattice-type fractals are Minkowski measurable.

• Theorem. A self-similar subset F ⊆ R is Minkowski measurable if

F is nonlattice. (Proved by Gatzouras.) The converse has kept us busy.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Converting “scales” to “sizes”

• Definition. The inradius of A ⊆ Rd is the radius of the largest

metric ball contained in A. Equivalently, ρ(A) := inf{ε > 0 .

. . A−ε = A}.

For A = G, write g := ρ(G). The tile Φw(G) has inradius rwg.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The scaling data of the tiling

Converting “scales” to “sizes”

• Definition. The inradius of A ⊆ Rd is the radius of the largest

metric ball contained in A. Equivalently, ρ(A) := inf{ε > 0 .

. . A−ε = A}.

For A = G, write g := ρ(G). The tile Φw(G) has inradius rwg. We now have all the scaling data we need. − → Next, we look at the geometry of the generator.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators

Monophase and pluriphase generators

• Definition. G is monophase iff VG(ε) is polynomial on [0, g]:

VG(ε) =

d−1

• k=0

κkεd−k, 0 ≤ ε ≤ g.

Motivation: Steiner’s Theorem. If A is convex, then vold(Aε) =

d−1

• k=0

κkεd−k for ε ≥ 0. Here, κk = µk(A) vold-k(Bd−k), and µk are the intrinsic volumes.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators

Monophase and pluriphase generators

• Definition. G is monophase iff VG(ε) is polynomial on [0, g]:

VG(ε) =

d−1

• k=0

κkεd−k, 0 ≤ ε ≤ g.

• Definition. G is pluriphase iff VG(ε) is polynomial on each

subinterval of [0 = a0, a1, a1, . . . , aM = g].

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators

How to compute the tube formula

Suppose you have Φ satisfying OSC, with int F = ∅, and a feasible open set O satisfying the compatibility theorem.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The geometric data of the generators

How to compute the tube formula

Suppose you have Φ satisfying OSC, with int F = ∅, and a feasible open set O satisfying the compatibility theorem. Scaling data: ζs(s) =

• w∈W

rs

w.

Generator geometry: VG(ε) =

d−1

• k=0

κkεd−k. We compute VT (ε) = vold(T−ε), the inner tube formula for the tiling, in terms of these ingredients.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula The fractal tube formula

Fractal tube formula

Theorem [Pearse 2006], [Lapidus, Pearse, Winter 2011] Suppose T has a single monophase generator and ζs has only simple poles. Then for 0 ≤ ε ≤ g, VT (ε) =

• ω∈Ds∪{0,1,...,d−1}

cωεd−ω, where cω =

• res (ζs; ω) d

k=0 gω−k ω−k κk,

ω ∈ Ds, κkζs(k), ω = k ∈ {0, 1, . . . , d}. Recall Ds := {poles of ζs}.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Lattice type fractals are not Minkowski measurable

Theorem [Lapidus, Pearse, Winter 2013]. Suppose F is a fractal with one monophase generator, d − 1 < D < d, and satisfying some technical conditions. If F is lattice, then F is not Minkowski measurable.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Lattice type fractals are not Minkowski measurable

Theorem [Lapidus, Pearse, Winter 2013]. Suppose F is a fractal with one monophase generator, d − 1 < D < d, and satisfying some technical conditions. If F is lattice, then F is not Minkowski measurable. Proof uses the pointwise tube formula, with an estimate based on careful choice of a window (contour).

D Wn+1 Wn



Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

The screen and the window

Choose a vertical contour S = {S(t) + ✐t .

. . t ∈ R}

where S : R → R is Lipschitz. S is the screen. Define the window W = {z ∈ C .

. . Re z ≥ S(Im z)}.

 D W 

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Fractal tube formula with error term

Theorem [Lapidus, Pearse, Winter 2013] Suppose T has a single monophase generator and ζs has only simple poles. Then for 0 ≤ ε ≤ g, VT (ε) =

• ω∈Ds∩W

cωεd−ω +

• k∈{0,1,...,d−1}∩W

ckεd−k + R(ε), where Ds := {poles of ζs} and cω = res (ζs; ω)

d

• k=0

gω−k ω − kκk, ck = κkζs(k), as before,

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Fractal tube formula with error term

Theorem [Lapidus, Pearse, Winter 2013] Suppose T has a single monophase generator and ζs has only simple poles. Then for 0 ≤ ε ≤ g, VT (ε) =

• ω∈Ds∩W

cωεd−ω +

• k∈{0,1,...,d−1}∩W

ckεd−k + R(ε), where Ds := {poles of ζs} and cω = res (ζs; ω)

d

• k=0

gω−k ω − kκk, ck = κkζs(k), as before, but now also R(ε) = 1 2π✐

• S

εd−sζs(s) d − s d−1

• k=0

gs−k s − k(d − k)κk(G)

• ds,

and R(ε) = O(εd−sup S), as ε → 0+.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Recent results

Theorem [Kombrink, Pearse, Winter 2014]. Suppose F is a fractal with pluriphase generators. If F is lattice, then F is not Minkowski measurable.

Pluriphase means generator has a piecewise polynomial inner tube formula.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Recent results

Theorem [Kombrink, Pearse, Winter 2014]. Suppose F is a fractal with pluriphase generators. If F is lattice, then F is not Minkowski measurable.

Pluriphase means generator has a piecewise polynomial inner tube formula.

Improvements on prior result: Compatibility (bd O ⊆ F) no longer required. Works for countably many generators. No longer require d − 1 < D < d. More elementary tools (“simpler” proof).

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

More precisely ...

• Theorem. F is the attractor of a lattice IFS satisfying OSC, and

D = dimM F. Also, O is a strongly feasible open set satisfying a projection condition, for which the generator is pluriphase. Strongly feasible means F ∩ O = ∅ (SOSC). Projection condition means SjO ⊆ π−1

F (SjF).

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

More precisely ...

• Theorem. F is the attractor of a lattice IFS satisfying OSC, and

D = dimM F. Also, O is a strongly feasible open set satisfying a projection condition, for which the generator is pluriphase.

1 If D = dim aff F, then F is Minkowski measurable.

In particular, this is true for D = d.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

More precisely ...

• Theorem. F is the attractor of a lattice IFS satisfying OSC, and

D = dimM F. Also, O is a strongly feasible open set satisfying a projection condition, for which the generator is pluriphase.

1 If D = dim aff F, then F is Minkowski measurable.

In particular, this is true for D = d.

2 If D /

∈ N, then F is not Minkowski measurable.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

More precisely ...

• Theorem. F is the attractor of a lattice IFS satisfying OSC, and

D = dimM F. Also, O is a strongly feasible open set satisfying a projection condition, for which the generator is pluriphase.

1 If D = dim aff F, then F is Minkowski measurable.

In particular, this is true for D = d.

2 If D /

∈ N, then F is not Minkowski measurable.

3 If D ∈ N, then F is not Minkowski measurable ⇐

⇒ certain algebraic relations involving the data of the pluriphase representation are satisfied. Lm(ε) =

• logr

am ε

• 1

28 1 14 1 7 2 7 4 7 4 7 3 28 3 14 3 7 6 7

• 5
• 4
• 3
• 2
• 1

1 2

L2(e) L1(e) = a2 g = a1

1 2

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

More precisely ...

• Theorem. F is the attractor of a lattice IFS satisfying OSC, and

D = dimM F. Also, O is a strongly feasible open set satisfying a projection condition, for which the generator is pluriphase.

1 If D = dim aff F, then F is Minkowski measurable.

In particular, this is true for D = d.

2 If D /

∈ N, then F is not Minkowski measurable.

3 If D ∈ N, then F is not Minkowski measurable ⇐

⇒ certain algebraic relations involving the data of the pluriphase representation are satisfied. Proof uses renewal theory.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Renewal Theorem. Suppose p1, . . . , pJ > 0 with J

j=1 pj = 1.

Let z : R → R have a discrete set of discontinuities and satisfy |z(t)| ≤ c1e−c2|t|, for some 0 < c1, c2 < ∞.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Renewal Theorem. Suppose p1, . . . , pJ > 0 with J

j=1 pj = 1.

Let z : R → R have a discrete set of discontinuities and satisfy |z(t)| ≤ c1e−c2|t|, for some 0 < c1, c2 < ∞. Let y1, . . . , yn > 0, and let Z be the unique solution of Z(t) = z(t) + J

j=1 pjZ(t − yj).

← (Renewal Eqn)

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Renewal Theorem. Suppose p1, . . . , pJ > 0 with J

j=1 pj = 1.

Let z : R → R have a discrete set of discontinuities and satisfy |z(t)| ≤ c1e−c2|t|, for some 0 < c1, c2 < ∞. Let y1, . . . , yn > 0, and let Z be the unique solution of Z(t) = z(t) + J

j=1 pjZ(t − yj).

← (Renewal Eqn) Then Z(t) has one of two asymptotic behaviors: (1) If {y1, . . . , yn} ⊆ hZ, then Z(t) ∼ h η

• k∈Z

z(t − kh), t → ∞. (2) If there is no such h, then lim

t→∞ Z(t) = 1

η ∞

−∞

z(τ) dτ. Here, η = J

j=1 pjyj.

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Tube formula as renewal process

Since s = D is the solution of rs

1 + · · · + rs n = 1, let pj = rD j .

V

(e)

rdV

(r-1e)

v(e)

VT (ε) =

J

• j=1

rd

j VT (r−1 j

ε) + v(ε)

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Tube formula as renewal process

Since s = D is the solution of rs

1 + · · · + rs n = 1, let pj = rD j .

V

(e)

rdV

(r-1e)

v(e)

εD−dVT (ε) =

J

• j=1

rD

j VT (r−1 j

ε)

• r−1

j

ε D−d + v(ε)εD−d

Minkowski dimension and nonmeasurability of lattice-type fractals Computation of the tube formula Minkowski (non)measurability results

Tube formula as renewal process

Since s = D is the solution of rs

1 + · · · + rs n = 1, let pj = rD j .

V

(e)

rdV

(r-1e)

v(e)

εD−dVT (ε) =

J

• j=1

rD

j VT (r−1 j

ε)

• r−1

j

ε D−d + v(ε)εD−d Z(ε) =

J

• j=1

pjZ(r−1

j

ε) + z(ε)

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

The distance and tube zeta functions

For a bounded set A ⊆ Rd, define the distance zeta function ζA(ε, s) :=

• Aε

dist(x, A)s−d dx, where Aε := {x ∈ Rd .

. . dist(x, A) ≤ ε} is the outer ε-neighbourhood

and dx is d-dim Lebesgue measure.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

The distance and tube zeta functions

For a bounded set A ⊆ Rd, define the distance zeta function ζA(ε, s) :=

• Aε

dist(x, A)s−d dx, where Aε := {x ∈ Rd .

. . dist(x, A) ≤ ε} is the outer ε-neighbourhood

and dx is d-dim Lebesgue measure. The tube zeta function is the Mellin transform of ε → vold(Aε):

• ζA(ε, s) :=

ε ts−d−1 vold(At) dt,

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

The distance and tube zeta functions

For a bounded set A ⊆ Rd, define the distance zeta function ζA(ε, s) :=

• Aε

dist(x, A)s−d dx, where Aε := {x ∈ Rd .

. . dist(x, A) ≤ ε} is the outer ε-neighbourhood

and dx is d-dim Lebesgue measure. The tube zeta function is the Mellin transform of ε → vold(Aε):

• ζA(ε, s) :=

ε ts−d−1 vold(At) dt, Easy theorem: ζA(ε, s) = εs−d vold(Aε) + (d − s) ζA(ε, s)

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

Fractal zeta functions: a different approach

ζA(ε, s) =

• Aε dist(x, A)s−d dx does not coincide with ζT for any A.

(e.g., for A = T = Φw(Gq)). However, it appears to contain comparable information. Theorem (Lapidus, Radunovic, Zubrinic 2011). dimM A = absc(ζA).

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

Fractal zeta functions: a different approach

ζA(ε, s) =

• Aε dist(x, A)s−d dx does not coincide with ζT for any A.

(e.g., for A = T = Φw(Gq)). However, it appears to contain comparable information. Theorem (Lapidus, Radunovic, Zubrinic 2011). dimM A = absc(ζA). Theorem (Kombrink, Pearse, Rock (last weekend)). For self-similar fractals F ⊆ Rd, the structure theorem for the complex dimensions of ζF is the same as for the scaling zeta function.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

Let O ⊆ Rd be open with λd(O) < ∞ and let A ⊆ Rd (poss. unbdd). The relative distance zeta function of A with respect to O is ζA(ε, s; O) =

• O

dist(x, A)s−d dx.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

Let O ⊆ Rd be open with λd(O) < ∞ and let A ⊆ Rd (poss. unbdd). The relative distance zeta function of A with respect to O is ζA(ε, s; O) =

• O

dist(x, A)s−d dx. The relative tube zeta function of A with respect to O is

• ζA(ε, s; O) =

ε ts−d−1 vold(At ∩ O) dt.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

• Theorem. Let F ⊆ Rd be a self-similar set satisfying OSC and let

O be a strongly feasible open set satisfying the projection

• condition. Then
• ζF(ε, s; O) =
• ζF(ε, s; Γ) + J

j=1 rs j

r−1

j

ε ε

ts−d−1 vold(Ft ∩ O) dt 1 − J

j=1 rs j

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

• Theorem. Let F ⊆ Rd be a self-similar set satisfying OSC and let

O be a strongly feasible open set satisfying the projection

• condition. Then
• ζF(ε, s; O) =
• ζF(ε, s; Γ) + J

j=1 rs j

r−1

j

ε ε

ts−d−1 vold(Ft ∩ O) dt 1 − J

j=1 rs j

Furthermore,

• ζF(ε, s) =

ζF(ε, s; O) + ζF(ε, s; Oc).

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

• Theorem. Let F ⊆ Rd be a self-similar set satisfying OSC and let

O be a strongly feasible open set satisfying the projection

• condition. Then
• ζF(ε, s; O) =
• ζF(ε, s; Γ) + J

j=1 rs j

r−1

j

ε ε

ts−d−1 vold(Ft ∩ O) dt 1 − J

j=1 rs j

Furthermore,

• ζF(ε, s) =

ζF(ε, s; O) + ζF(ε, s; Oc).

• Corollary. The structure theorem for the complex dimensions of

the tube and distance zeta functions is the same as for the scaling zeta function.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions A new hope; a new beginning

A little more detail ...

• Theorem. Let F ⊆ Rd be a self-similar set satisfying OSC and let

O be a strongly feasible open set satisfying the projection

• condition. Then
• ζF(ε, s; O) =
• ζF(ε, s; Γ) + J

j=1 rs j

r−1

j

ε ε

ts−d−1 vold(Ft ∩ O) dt 1 − J

j=1 rs j

Furthermore,

• ζF(ε, s) =

ζF(ε, s; O) + ζF(ε, s; Oc).

• Corollary. The structure theorem for the complex dimensions of

the tube and distance zeta functions is the same as for the scaling zeta function.

• Corollary. For F ⊆ R, the lattice/nonlattice dichotomy extends to

the distance and tube zeta functions.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The Steiner formula: a tube formula for convex bodies

Let A ∈ Kd be a convex body (nonempty, convex, compact set). vold(A + εBd) =

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The Steiner formula: a tube formula for convex bodies

Let A ∈ Kd be a convex body (nonempty, convex, compact set). vold(A + εBd) = c0ε2 + . . .

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The Steiner formula: a tube formula for convex bodies

Let A ∈ Kd be a convex body (nonempty, convex, compact set). vold(A + εBd) = c0ε2 + c1ε1 + . . .

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The Steiner formula: a tube formula for convex bodies

Let A ∈ Kd be a convex body (nonempty, convex, compact set). vold(A + εBd) = c0ε2 + c1ε1 + c2ε0 = d

i=0 ci εd−i.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The coefficients of the Steiner formula describe curvature

The coefficients ci = d

i

• Ci(A) are interesting:

Ci is the ith total curvature of A. C0 is the Euler characteristic C1 is mean width Cd−1 is surface area Cd is volume Ci(A) is defined via integral geometry in terms of average volumes

• f projections to lower-dimensional subspaces.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The Steiner formula

The coefficients of the Steiner formula describe curvature

The coefficients ci = d

i

• Ci(A) are interesting:

Ci is the ith total curvature of A. C0 is the Euler characteristic C1 is mean width Cd−1 is surface area Cd is volume Also, ci = ωd−iµi(A), where µi is the ith intrinsic volume of A and ωi = voli(Bi). Ci is homogeneous of degree i: Ci(λA) = λiCi(A), λ > 0.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

The coefficients Ci can be localized as measures

Ci is total i-dimensional curvature measure. Localization: Ci(A, ·) is a measure defined on B(Rd). For B ⊆ Rd, Ci(A, B) describes curvature of B ∩ ∂A.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

The coefficients Ci can be localized as measures

Ci is total i-dimensional curvature measure. Localization: Ci(A, ·) is a measure defined on B(Rd). For B ⊆ Rd, Ci(A, B) describes curvature of B ∩ ∂A. R = {x ∈ Aε .

. . p(A, x) ∈ B}. VA(ε, B) = d−1 i=0 Ci(A, B)εd−i = vol2(R).

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

The coefficients Ci can be localized as measures

Ci is total i-dimensional curvature measure. Localization: Ci(A, ·) is a measure defined on B(Rd). For B ⊆ Rd, Ci(A, B) describes curvature of B ∩ ∂A. R = {x ∈ Aε .

. . p(A, x) ∈ B}. VA(ε, B) = d−1 i=0 Ci(A, B)εd−i = vol2(R).

C1(A, B) > 0, but C0(A, B) = 0. (VA(ε, B) grows linearly in ε)

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

The coefficients Ci can be localized as measures

Ci is total i-dimensional curvature measure. Localization: Ci(A, ·) is a measure defined on B(Rd). For B ⊆ Rd, Ci(A, B) describes curvature of B ∩ ∂A.

R′ = {x ∈ Aε .

. . p(A, x) ∈ B′}. VA(ε, B′) = d−1 i=0 Ci(A, B′)εd−i = vol2(R′).

Now C1(A, B′) > 0 and C0(A, B′) > 0.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

Additive extension of the Steiner formula

A is polyconvex iff A is a finite union of convex sets. A ∈ U(Kd) iff A = N

n=1[An],

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

Additive extension of the Steiner formula

A is polyconvex iff A is a finite union of convex sets. A ∈ U(Kd) iff A = N

n=1[An],

The curvature measures are valuations. If A1, A2, A1 ∪ A2 ∈ Kd, then Ci(A1 ∪ A2, ·) + Ci(A1 ∩ A2, ·) = Ci(A1, ·) + Ci(A2, ·). Inclusion-exclusion principle can be used to define Ci(A, ·) for A ∈ U(Kd).

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

Additive extension of the Steiner formula

Summary: If A ∈ U(Rd) is a polyconvex set, then∗ VA(ε, B) = d−1

i=0 Ci(A, B)εd−i.

More conceptually: VA(ε) =

• i∈{0,...,d−1} ciεd−i,

where i = 0, . . . , d − 1 are “integer dimensions” of A, the coefficients ci are related to curvature.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

ζA may lend itself more easily to localization

For Σ := Rd × Sd−1, consider a Borel subset η ∈ B(Σ). For A convex, let p(A, ·) be metric projection to A. Let u(A, x) denote the unit vector pointing from p(A, x) to x. A S1

x p(A,x) u(A,x)

The normal bundle of A is Nor A := {(p(A, x), u(A, x)) .

. . x ∈ Rd}.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

A local parallel set is MA(ε, η) := {x ∈ Rd .

. . 0 < dist(x, A) ≤ ε and (p(A, x), u(A, x)) ∈ η}.

A A β η=β×σ MA(ε,η) σ

Taking η = β × Sd−1 recovers the earlier case.

Minkowski dimension and nonmeasurability of lattice-type fractals Fractal zeta functions The curvature measures

A local parallel set is MA(ε, η) := {x ∈ Rd .

. . 0 < dist(x, A) ≤ ε and (p(A, x), u(A, x)) ∈ η}.

For a bounded set A ⊆ Rd, define ζA(ε, s; η) :=

• MA(ε,η)

dist(x, A)s−d dx. This approach builds in localization from the beginning.