Fractal Geometry, Complex Dimensions and Zeta Functions

Lapidus, Michel L.; Frankenhuijsen, Machiel van

doi:10.1007/978-0-387-35208-4

Cited by 131 publications

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“…The proofs extend without significant difficulty to vectors γ ∈ (S 1 ) r . Parts (iv), (v) of Lemma 1 extend the discussion in [20] (see Theorems 3.25 and 3.26) since this only seems to apply to γ ∈ N r . The reader may also find it helpful to consult the article [8] that adds additional and useful details to those given in [20].…”

Section: Two Technical Preliminariessupporting

confidence: 52%

“…The first point concerns the proof of Part (iii). This is an immediate consequence of Theorem 3.6 of [20].…”

Section: Two Technical Preliminariesmentioning

confidence: 56%

“…As a result, we see that the general case reduces to the particular case where γ j = 1 for all j, to which Theorems 3.25, 3.26 of [20] do apply. This completes the proof of the lemma.…”

Section: Proof Of Partmentioning

confidence: 84%

“…We first recall from [19,20] that there are two distinct cases 'latticelike case' and 'nonlatticelike case'. This depends solely upon the behavior of the roots of the polynomial Λ(c, 1, s).…”

Section: Two Technical Preliminariesmentioning

confidence: 99%

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Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

Essouabri¹,

Lichtin²

2015

Proc. London Math. Soc.

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This paper uses the zeta function methods to solve Falconer-type problems about sets of k-simplices whose endpoints belong simultaneously to a self-similar subset F of Z n (k n) and a disc B(x) of a large radius x. Assuming that the similarity transformations pairwise commute, we study four Euclidean metric invariants of the simplices, the most basic (and frequently studied) of which is the distance between endpoints of a 1-simplex. For each, we introduce a zeta function, derive its functional properties, and apply such information to derive a lower bound on the upper Minkowski dimension of F , which guarantees that the number of distinct metric invariants must be unbounded as x → ∞.

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Section: Two Technical Preliminariessupporting

confidence: 52%

“…The first point concerns the proof of Part (iii). This is an immediate consequence of Theorem 3.6 of [20].…”

Section: Two Technical Preliminariesmentioning

confidence: 56%

“…As a result, we see that the general case reduces to the particular case where γ j = 1 for all j, to which Theorems 3.25, 3.26 of [20] do apply. This completes the proof of the lemma.…”

Section: Proof Of Partmentioning

confidence: 84%

Section: Two Technical Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

Essouabri¹,

Lichtin²

2015

Proc. London Math. Soc.

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“…Lapidus and Van Frankenhuysen [1][2][3] consider the functions known as nonlattice Dirichlet polynomials, which are exponential polynomials of the form…”

Section: Introductionmentioning

confidence: 99%

On the existence of exponential polynomials with prefixed gaps

Mora¹,

Sepulcre²,

Vidal³

2013

Bulletin of the London Mathematical Society

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This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P (z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP , the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

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Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

Park,

Xiao

2023

Mathematische Nachrichten

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We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in , , with respect to subordinate killed Brownian motions via ‐stable subordinators and establish the asymptotic behavior of the spectral heat content as for the full range of . Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when , where is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

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Fractal Geometry, Complex Dimensions and Zeta Functions

Cited by 131 publications

References 0 publications

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

On the existence of exponential polynomials with prefixed gaps

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

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Fractal Geometry, Complex Dimensions and Zeta Functions

Cited by 131 publications

References 0 publications

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤn

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤn

On the existence of exponential polynomials with prefixed gaps

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

Contact Info

Product

Resources

About

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤⁿ