Tube Zeta Function in Fractal Geometry

• Tube zeta functions are defined via Mellin transforms of tubular neighborhood volumes, linking Minkowski dimension and measurable fractal geometry.
• They connect with distance zeta functions through a functional identity, allowing detailed residue analysis to extract complex dimensions and oscillatory behavior.
• The framework extends to relative fractal drums, enabling study of nontraditional geometries and offering precise tube formulas that refine Minkowski content.

Searching arXiv for recent and foundational papers on tube zeta functions and related fractal zeta functions. The tube zeta function is a fractal zeta function associated with the volume growth of tubular neighborhoods of a set or, more generally, of a relative fractal drum. For a bounded set $A\subset\mathbb{R}^N$ and fixed $\delta>0$, it is defined by

$$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$

where $A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$ and $|A_t|$ is the $N$-dimensional Lebesgue measure of the $t$-neighborhood of $A$ (Lapidus et al., 2015). In the relative setting, for a relative fractal drum $(A,\Omega)$, the corresponding function is

$$\widetilde{\zeta}_{A,\Omega}(s;\delta):=\int_0^{\delta} t^{\,s-N-1}\, |A_t \cap \Omega|_N \, dt,$$

with $\delta>0$0 the relative tube volume (Lapidus et al., 2016). This construction treats the tube function $\delta>0$1 or $\delta>0$2 as a Mellin-type object, so that geometric scaling exponents, oscillations, and Minkowski content become accessible through the poles and residues of a meromorphic continuation. In the literature of Lapidus, Radunović, and Žubrinić, these poles are the complex dimensions of the underlying set or drum (Lapidus et al., 2014).

1. Definition and basic geometric setting

For a bounded subset $\delta>0$3, the tube function is the map

$\delta>0$4

The tube zeta function is then

$\delta>0$5

initially for $\delta>0$6 sufficiently large (Lapidus et al., 2015). The power $\delta>0$7 is chosen so that if $\delta>0$8 as $\delta>0$9, then the integral behaves like $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$0, making $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$1 the natural singularity (Lapidus et al., 2015).

The relative theory replaces $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$2 by an ordered pair $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$3, called a relative fractal drum (RFD), where $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$4 is arbitrary, $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$5 is Lebesgue measurable with finite volume, and $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$6 for some $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$7 (Lapidus et al., 2016). The relative tube zeta function is

$$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$8

and it generalizes both bounded sets and fractal strings (Lapidus et al., 2015).

A recurring point in the theory is that the choice of $$\widetilde\zeta_A(s):=\int_0^\delta t^{\,s-N-1}|A_t|\,dt,$$9 affects the function only by an entire term, so the poles and residues relevant for complex dimensions are independent of $A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$0 (Lapidus et al., 2015). This makes the tube zeta function a local geometric invariant of the small-$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$1 asymptotics of tubular neighborhoods.

2. Relation to distance zeta functions and Mellin analysis

The tube zeta function is closely related to the distance zeta function

$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$2

and, in the relative case,

$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$3

(Lapidus et al., 2015). The central functional identity is

$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$4

valid on the initial half-plane of convergence and, by continuation, on any domain where either side is meromorphic (Lapidus et al., 2016).

This identity has several immediate consequences. First, the distance and tube zeta functions carry essentially the same information, except possibly at $A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$5. Second, their poles coincide away from $A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$6, and for a simple pole $A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$7,

$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$8

(Lapidus et al., 2014). Third, the tube zeta function is more directly tied to the tube geometry because it is essentially the Mellin transform of the modified tube function

$A_t=\{x\in\mathbb{R}^N:d(x,A)<t\}$9

(Lapidus et al., 2016).

This Mellin-transform viewpoint underlies the derivation of tube formulas. Mellin inversion yields

$|A_t|$0

and contour shifting then converts this inverse transform into a sum of residues over the poles of $|A_t|$1 (Lapidus et al., 2016). This is the analytic mechanism by which the complex dimensions determine tube asymptotics.

3. Box dimension, residues, and Minkowski content

A foundational result is that the abscissa of absolute convergence of the tube zeta function equals the upper box dimension. For bounded $|A_t|$2,

$|A_t|$3

and analogously for RFDs,

$|A_t|$4

(Lapidus et al., 2015). Thus the critical line $|A_t|$5 in the complex plane corresponds to the geometric scaling threshold.

Under Minkowski nondegeneracy assumptions, the principal pole at $|A_t|$6 is simple and its residue is controlled by the lower and upper Minkowski contents. For the relative tube zeta function, if $|A_t|$7 exists and $|A_t|$8 is Minkowski nondegenerate, then

$|A_t|$9

and in the Minkowski measurable case,

$N$0

(Lapidus et al., 2016). The bounded-set version is stated in the same form in (Lapidus et al., 2015, Lapidus et al., 2015).

This residue formula is one of the main reasons the tube zeta function is central: it refines the bare value of the Minkowski dimension into a complex-analytic invariant whose principal residue recovers Minkowski content. In the measurable case, the leading singular behavior of $N$1 near $N$2 exactly matches the asymptotic law

$N$3

A related criterion links Minkowski measurability to the pole structure on the critical line. Under suitable admissibility and growth hypotheses, an RFD is Minkowski measurable if and only if $N$4 is the only pole on $N$5 and that pole is simple (Lapidus et al., 2015). This excludes leading-order oscillatory poles on the critical line in the measurable case.

4. Complex dimensions and fractal tube formulas

The poles of a meromorphic continuation of the tube zeta function are the complex dimensions. More precisely, if $N$6 extends meromorphically to a connected neighborhood $N$7 of the critical line, then the visible complex dimensions relative to $N$8 are the poles of that extension in $N$9, and the principal complex dimensions are those with real part equal to $t$0 (Lapidus et al., 2016).

Under suitable languidity conditions, the tube volume admits a residue expansion. In the simple-pole case, the pointwise fractal tube formula is

$t$1

where the sum runs over visible complex dimensions in a window $t$2 and the remainder comes from integration over the screen (Lapidus et al., 2016). If the tube zeta function is strongly languid, the contour can be pushed arbitrarily far left and the error term vanishes, producing an exact tube formula (Lapidus et al., 2016).

The interpretation of an individual term

$t$3

is geometrically significant. The real part $t$4 determines the order of magnitude, while the imaginary part $t$5 determines log-periodic oscillation frequency. In this sense, nonreal complex dimensions are analytic markers of oscillatory geometry (Lapidus et al., 2015).

Multiple poles lead to logarithmic corrections. If $t$6 is a pole of multiplicity $t$7, then the corresponding contribution to the tube formula involves

$t$8

(Lapidus et al., 2015). Such logarithmic factors occur in more intricate scaling situations and in higher-dimensional spray constructions.

5. Principal examples and model geometries

Several canonical examples illustrate the range of the theory.

5.1 Fractal strings and generalized Cantor sets

In one dimension, the geometric zeta function of a fractal string becomes a special case of the distance zeta function, hence also of the tube zeta framework (Lapidus et al., 2016). For generalized Cantor sets $t$9, one has

$A$0

and the tube function takes the form

$A$1

where $A$2 is periodic with period $A$3 (Lapidus et al., 2015). This yields a lattice of principal complex dimensions

$A$4

(Lapidus et al., 2015). These are the prototypical log-periodic oscillatory complex dimensions.

5.2 Sierpiński gasket and Sierpiński carpet

For the planar Sierpiński gasket, the distance zeta function is explicitly computable and has complex dimensions

$A$5

all simple (Lapidus et al., 2016). The corresponding tube formula contains an oscillatory leading term of order $A$6, together with integer-dimensional terms.

For the Sierpiński carpet, the tube zeta picture is analogous: the critical line contains the vertical lattice

$A$7

reflecting self-similar oscillations (Lapidus et al., 2015).

5.3 Smooth sets

The theory also recovers classical smooth geometry. For a sphere $A$8, the tube zeta function extends meromorphically with only finitely many real poles, and the principal complex dimension is $A$9 (Lapidus et al., 2015). This shows that nonreal complex dimensions are not generic: they are associated with oscillatory fractal scaling rather than mere nonsmoothness.

5.4 Relative examples and negative dimensions

Relative fractal drums enlarge the theory beyond bounded sets and permit negative box dimensions. For the RFD with $(A,\Omega)$0 and

$(A,\Omega)$1

the relative box dimension is

$(A,\Omega)$2

(Lapidus et al., 2014). This shows that tube zeta functions are not confined to positive-dimensional fractal geometry. They also detect extremely flat relative configurations.

6. Generalizations, variants, and conceptual scope

The term “tube zeta function” appears in several related but distinct settings. The main Lapidus–Radunović–Žubrinić theory concerns bounded sets and relative fractal drums in Euclidean spaces (Lapidus et al., 2015, Lapidus et al., 2015, Lapidus et al., 2016, Lapidus et al., 2014). A related but different construction appears for fractal sprays and self-similar tilings, where the relevant object is the tubular zeta function

$(A,\Omega)$3

built from the scaling zeta function of the underlying fractal string together with Steiner-like data of the generator (Lapidus et al., 2010). Its poles are the poles of the scaling zeta function together with the integers $(A,\Omega)$4, and its residues yield pointwise tube formulas for sprays and self-similar tilings (Lapidus et al., 2010).

Another extension occurs in dynamics. For an orbit $(A,\Omega)$5 of a parabolic germ, the relative tube zeta function is

$(A,\Omega)$6

where $(A,\Omega)$7 is the tube function of the orbit (Mardešić et al., 2020). In that setting the zeta function extends meromorphically to all of $(A,\Omega)$8, but the resulting complex dimensions are all real. This is a notable counterpoint to self-similar fractals: higher-order oscillations in the tube function need not produce nonreal complex dimensions (Mardešić et al., 2020).

The phrase “Tube Zeta Function” should not be confused with unrelated uses of “tube” in complex analysis. For example, “tube domains” in the theory of theta functions concern domains of the form $(A,\Omega)$9 and have no direct relation to fractal tube neighborhoods (Dorfmeister et al., 2015). Likewise, the “Tube Zeta Function” is unrelated to the Riemann–Siegel $$\widetilde{\zeta}_{A,\Omega}(s;\delta):=\int_0^{\delta} t^{\,s-N-1}\, |A_t \cap \Omega|_N \, dt,$$0-function despite occasional lexical similarity with “tube” or “strip” language in other contexts (Reyna, 2024). This distinction is important because the object treated in fractal geometry is explicitly defined from tubular neighborhood volumes, not from analytic continuation on tube domains or strip kernels.

A broader implication of the theory is that the tube zeta function functions as an analytic encoding of small-scale geometry. It connects tube asymptotics, Minkowski content, oscillatory structure, and residue calculus in a single framework. In the strongest cases, such as strongly languid self-similar constructions, it yields exact tube formulas (Lapidus et al., 2016). In more singular cases, such as maximally hyperfractal sets, it can exhibit a singularity at every point of the critical line, producing natural boundaries and extremely dense oscillatory spectra (Lapidus et al., 2015). This suggests that the tube zeta function is not merely a reformulation of Minkowski dimension, but a higher-resolution invariant of geometric complexity.