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 it . . . 8 1/81 x 4 1/27 x 2 x 1/9 1/3 σ 1 0 1/2 log 2 3 “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 ∞ n s = 1 + 1 1 2 s + 1 3 s + 1 4 s + 1 5 s + 1 6 s + 1 7 s + 1 8 s + 1 � ζ ( s ) = 9 s + · · · n =1 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 it . . . 8 1/81 x 4 1/27 x 2 x 1/9 1/3 σ 1 0 1/2 log 2 3 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 oscillations 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 . . . 8 x 1/81 4 1/27 x 2 x 1/9 . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1/3 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 ⊂ R m are given, respectively, by vol m ( A ε ) vol m ( A ε ) M r ∗ ( A ) = lim sup M r , ∗ ( A ) = lim inf . (1) ε m − r ε m − r ε → 0 + ε → 0 + The upper and lower Minkowski dimensions of a bounded set A are defined by dim M A = inf { r ≥ 0 : M r ∗ ( A ) = 0 } = sup { r ≥ 0 : M r ∗ ( A ) = ∞} , (2) dim M A = inf { r ≥ 0 : M r ∗ ( A ) = 0 } = sup { r ≥ 0 : M r ∗ ( A ) = ∞} . (3) If dim M A = dim M A , the common value is called the Minkowski dimension of A and is denoted dim M A . John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 7 / 79

Minkowski content and measurability Definition Let A ⊂ R m be such that D M = dim M A exists. The upper and lower Minkowski contents of A are denoted as M ∗ ( A ) = M ∗ D M ( A ) , and (4) M ∗ ( A ) = M D M ( 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 vol m ( A ε ) M ( A ) = lim ε m − D M . (6) ε → 0 + 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 = ( ℓ j ) ∞ j =1 ), then   ∞   � ℓ t dim M ( ∂ Ω) = D L := inf  t ∈ R : j < ∞  , (7) j =1 where ∂ Ω = [ a, b ] \ Ω and dim M 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 = ( ℓ j ) ∞ “=” { l n : l n distinct with multiplicity m n , n ∈ N } (8) j =1 where 0 < ℓ j +1 ≤ ℓ j ∀ j and ℓ j → 0 . The dimension D L , geometric zeta function ζ L , and complex dimensions D L of L , are given by � � � ℓ t D L := inf t ∈ R : j < ∞ , (9) � � ℓ s m n l s ζ L ( s ) := j = n , (10) D L ( W ) := { ω ∈ W ⊂ C : ζ L has a pole at ω } , (11) where Re ( s ) > D L and W is a suitable open region. If W = C , we write D L for D L ( W ) . John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 10 / 79

The Cantor string it . . . 8 1/81 x 4 x 1/27 2 1/9 x 1/3 σ 0 1/2 log 2 1 3 D L CS = log 2 L CS = { 3 − n : multiplicity 2 n − 1 , n ∈ N } , log 3 , (12) ∞ 3 − s 2 n − 1 3 − ns = � ζ L CS = 1 − 2 · 3 − s , and (13) n =1 2 π D L CS = { D L CS + pki : k ∈ Z } , where p = log 3 . (14) John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 11 / 79

Geometric counting functions Figure : Plot of N L CS ( x ) /x D CS . (Think of Minkowski content as x → ∞ .) Definition (LvF) For x > 0 , the geometric counting function of a fractal string L is given by N L ( x ) := # { j ∈ N : ℓ − 1 � ≤ x } = m n . (15) j n ∈ N , l − 1 n ≤ x 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 N L . Then for Re ( s ) > D L we have � ∞ � m n l s N L ( x ) x − s − 1 dx. ζ L ( s ) = n = s (16) 0 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 D L ( W ) consists entirely of simple poles. Then, under certain growth conditions (see the next two slides) on ζ L , we have x ω � N L ( x ) = ω res ( ζ L ( s ); ω ) + { ζ L (0) } + R ( x ) , (17) ω ∈D L ( W ) where R ( x ) is an error term of small order and the term in braces is included only if 0 ∈ W \D L . For the Cantor string L CS we have x D + izp 1 N L CS ( x ) = 2 n − 1 = � D + izp − 1 (18) 2 log 3 z ∈ Z 2 π where D = log 3 2 , p = log 3 , and n = [log 3 x ] (integer part of log 3 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 ζ L satisfies the following conditions: There exist constants κ, C > 0 and a two-sided sequence { T z } z ∈ Z of real numbers where T − n < 0 < T n for n ≥ 1 , and T n n →∞ T n = ∞ , lim n →∞ T − n = −∞ , lim lim | T − n | = 1 (19) n → + ∞ such that for a suitable W and ‘screen’ S = ∂W we have L1 for all n ∈ Z and all σ ≥ S ( T n ) , | ζ L ( σ + iT n ) | ≤ C · ( | T n | + 1) κ , (20) L2 for all t ∈ R , | t | ≥ 1 , | ζ L ( 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)), ζ L satisfies the following condition: There exists a sequence of screens S m : t �→ S m ( t ) + it for m ≥ 1 , t ∈ R , with sup S m → −∞ as m → ∞ and with a uniform Lipschitz bound sup m ≥ 1 � S m � Lip < ∞ , such that L2’ There exist constants A, C > 0 such that for all t ∈ R and m ≥ 1 , | ζ L ( S m ( t ) + it ) | ≤ CA | S m ( 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 = D L , and it is simple. 2 N L ( x ) = cx D L + o ( x D L ) 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 it . . . 8 1/81 x 4 1/27 x 2 1/9 x 1/3 σ log 2 1 0 1/2 3 John A. Rock (Cal Poly Pomona) Dimensions, Measurability, Dichotomies 2015.03.31 18 / 79