Distance Zeta Function in Fractal Geometry

• Distance zeta function is a complex-analytic tool that encodes the geometric features of bounded sets using the distribution of Euclidean distances.
• It captures the Minkowski (box) dimension and oscillatory scaling laws by identifying poles through analytic continuation.
• It extends classical fractal string theory to higher dimensions and generalizes to relative fractal drums, probability measures, and noncommutative settings.

Searching arXiv for recent and foundational papers on distance zeta functions. The distance zeta function is a complex-analytic invariant associated with a bounded set $A\subset \mathbb{R}^N$. It encodes the geometry of $A$ through the distribution of Euclidean distances to $A$ inside a tubular neighborhood, and its singularities are used to detect box or Minkowski dimension, Minkowski content, and oscillatory scaling phenomena. Introduced as a higher-dimensional extension of the geometric zeta functions of bounded fractal strings, it is defined for arbitrary bounded sets and later generalized to relative fractal drums, probability measures, and a class of states on a $C^*$-algebra (Lapidus et al., 2015, Lapidus et al., 2016, Chow, 26 Feb 2025).

1. Definition and basic analytic setup

Fix a bounded, nonempty set $A\subset \mathbb{R}^N$ and $\delta>0$. Its open $\delta$-neighborhood is

$A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$

where $d(x,A)$ denotes the Euclidean distance from $x$ to $A$0. The distance zeta function of $A$1 is

$A$2

for $A$3 with $A$4 sufficiently large. Here $A$5 is $A$6-dimensional Lebesgue measure, and the exponent $A$7 makes the integral sensitive to the local distribution of distances near $A$8 (Lapidus et al., 2015).

This construction is designed to work for any bounded set, not only for one-dimensional strings. A useful technical point is that $A$9 is a Dirichlet-type integral, more specifically one of the tamed Dirichlet-type integrals discussed in the theory. The boundedness assumption is essential in the stated framework, while compactness may be assumed without loss of generality because replacing $A$0 by $A$1 does not change the relevant quantities (Lapidus et al., 2015).

The dependence on $A$2 is not essential in the sense that changing $A$3 modifies $A$4 only by an entire function. Accordingly, the poles and the analytic behavior near the critical line are unaffected. This addresses a common misconception that the theory depends in an essential way on the arbitrary tubular radius: the local singular structure is stable under this change (Lapidus et al., 2015).

2. Convergence, holomorphy, and Minkowski dimension

A central theorem identifies the abscissa of absolute convergence of $A$5 with the upper box dimension of $A$6: $A$7 Equivalently, $A$8 is holomorphic in the half-plane

$A$9

and this half-plane is optimal for the defining integral. Thus the upper box dimension acquires an analytic characterization as a boundary of convergence (Lapidus et al., 2015).

Under additional mild assumptions, if the box dimension $C^*$0 exists and $C^*$1 is Minkowski nondegenerate with

$C^*$2

then the abscissa of holomorphic continuation also equals $C^*$3: $C^*$4 This links the analytic continuation problem to the geometry of tubular growth near the set (Lapidus et al., 2015).

The same principle persists in the broader relative framework. For a relative fractal drum $C^*$5, the relative distance zeta function is

$C^*$6

and its abscissa of absolute convergence is the relative upper box dimension: $C^*$7 If $C^*$8 exists and $C^*$9, then

$A\subset \mathbb{R}^N$0

This places ordinary bounded sets and relative objects inside a single dimensional-analytic formalism (Lapidus et al., 2016).

3. Complex dimensions and the critical line

If $A\subset \mathbb{R}^N$1 admits a meromorphic continuation to a neighborhood of the critical line

$A\subset \mathbb{R}^N$2

then the poles on that line are called the principal complex dimensions of $A\subset \mathbb{R}^N$3: $A\subset \mathbb{R}^N$4 More generally, relative to a chosen window $A\subset \mathbb{R}^N$5, the visible complex dimensions are the poles of the meromorphic continuation lying in that region (Lapidus et al., 2015).

These poles refine the information carried by the real dimension alone. The box dimension records a single threshold, whereas the principal complex dimensions encode oscillatory and arithmetic information through their placement on the critical line. A plausible implication is that two sets with the same box dimension can nevertheless exhibit different logarithmic oscillation patterns, reflected in different pole configurations.

In the theory of relative fractal drums, once $A\subset \mathbb{R}^N$6 or $A\subset \mathbb{R}^N$7 is meromorphically continued to a neighborhood of

$A\subset \mathbb{R}^N$8

its poles are the complex dimensions of the RFD, and the poles on the critical line are again the principal complex dimensions: $A\subset \mathbb{R}^N$9 This recovers the earlier theory of complex dimensions for fractal strings and extends it to compact sets, sprays, tilings, and related geometric objects (Lapidus et al., 2016).

A recurrent misconception is to identify the distance zeta function solely with the detection of a real-valued dimension. In the cited theory, the critical role of meromorphic continuation is precisely to reveal nonreal poles, and hence oscillatory scaling laws, beyond the abscissa of convergence.

4. Tube zeta function and residue formulas

Closely related to $\delta>0$0 is the tube zeta function

$\delta>0$1

where

$\delta>0$2

and $\delta>0$3 is the $\delta>0$4-dimensional volume of the $\delta>0$5-neighborhood. Since Minkowski content and box dimension are formulated through the asymptotics of $\delta>0$6, this tube-based form is especially natural in fractal geometry (Lapidus et al., 2015).

The two zeta functions are linked by the functional equation

$\delta>0$7

valid for $\delta>0$8 large enough and then by analytic continuation whenever both sides make sense. In particular, when $\delta>0$9, they have the same abscissa of convergence and essentially the same meromorphic continuation behavior near the critical line (Lapidus et al., 2015).

The residue at the box dimension recovers Minkowski content. Writing

$\delta$0

if $\delta$1 is Minkowski measurable, so that

$\delta$2

and if the relevant zeta function is meromorphic near $\delta$3, then

$\delta$4

The tube zeta function is therefore especially clean: its residue at the box dimension equals the Minkowski content itself (Lapidus et al., 2015).

Without Minkowski measurability, the residue remains constrained by the lower and upper Minkowski contents: $\delta$5 and

$\delta$6

The relative theory for RFDs has the analogous residue estimate

$\delta$7

with equality in the Minkowski measurable case (Lapidus et al., 2016).

5. Relative fractal drums and explicit examples

The theory of relative fractal drums generalizes distance zeta functions from bounded sets to ordered pairs $\delta$8, where $\delta$9 is open, possibly unbounded, but of finite Lebesgue measure, $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$0, and there exists $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$1 such that $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$2. The associated relative distance and tube zeta functions are

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

This framework contains bounded fractal strings and compact subsets of Euclidean space as special cases. In particular, for a suitable geometric realization of a bounded fractal string $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$4,

$A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$5

and for a bounded set $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$6,

$A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$7

The unification is one of the defining features of the theory (Lapidus et al., 2016).

Several examples exhibit explicit pole structures. For the generalized Cantor set $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$8, assuming $A_\delta:=\{x\in\mathbb{R}^N:d(x,A)<\delta\},$9,

$d(x,A)$0

and its meromorphic continuation has poles

$d(x,A)$1

This gives a periodic array of complex dimensions and makes explicit the relation between logarithmic periodicity and nonreal poles (Lapidus et al., 2016).

For the relative Sierpiński gasket,

$d(x,A)$2

so that

$d(x,A)$3

For the relative Sierpiński carpet,

$d(x,A)$4

and

$d(x,A)$5

These formulas display self-similar pole lattices in a particularly transparent form (Lapidus et al., 2016).

Smooth or piecewise smooth situations also fit the theory. For the sphere $d(x,A)$6 with $d(x,A)$7,

$d(x,A)$8

hence

$d(x,A)$9

More generally, for boundaries of compact sets of positive reach, Federer’s tube formula yields meromorphic zeta functions with poles at integer dimensions (Lapidus et al., 2016).

The framework also clarifies the Cantor graph or devil’s staircase example. For the associated RFD,

$x$0

so

$x$1

In this language, the graph is not critically fractal but strictly subcritically fractal, which is used to resolve the “devil’s staircase paradox” (Lapidus et al., 2016).

6. Quasiperiodicity, hyperfractality, and later extensions

The distance zeta function is particularly effective in describing oscillatory geometries. One family of examples is provided by transcendentally quasiperiodic sets, introduced using unions of generalized Cantor sets together with Baker’s theorem and the Gel’fond–Schneider theorem. In the finite-dimensional setting, these sets have tube asymptotics involving sums of periodic functions with incommensurable periods whose ratios are transcendental. The cited work shows how to build sets with any prescribed finite number of such quasiperiods (Lapidus et al., 2015).

The RFD memoir pushes this further to transcendentally $x$2-quasiperiodic RFDs, for which the oscillatory term $x$3 is built from infinitely many algebraically independent periods. It also introduces increasingly strong singularity notions: a hyperfractal has a natural boundary along some screen, a strong hyperfractal has the critical line $x$4 as a natural boundary, and a maximal hyperfractal has every point of that critical line as a nonremovable singularity. The authors construct RFDs with the entire critical line consisting of nonremovable singularities, as well as Minkowski measurable RFDs possessing an infinite sequence of complex dimensions of arbitrary multiplicity $x$5 and even an infinite sequence of essential singularities along the critical line (Lapidus et al., 2016).

A later extension generalizes the relative distance zeta function from sets to probability measures and then to a class of states on a $x$6-algebra. For a probability measure $x$7 on $x$8, with

$x$9

and another finite Borel measure $A$00, the relative distance zeta function is

$A$01

with relative complex dimensions defined as the poles of its meromorphic continuation. A measure is called fractal with respect to $A$02 if the set of relative complex dimensions has a nonreal element (Chow, 26 Feb 2025).

In the noncommutative setting, for a state $A$03 and a measure-like functional $A$04, the relative distance zeta functional is

$A$05

with corresponding tube zeta functional, relative Minkowski contents, relative box dimensions, and relative complex dimensions. It is well-defined and holomorphic for

$A$06

admits transformation rules under scaling, translation, orthogonal rotations, and certain inner automorphisms, and satisfies tensor product dimension formulas such as

$A$07

The paper also proposes self-similar functional equations of the form

$A$08

suggesting arithmetic progressions of poles in the imaginary direction, in direct analogy with classical self-similar fractals (Chow, 26 Feb 2025).

Across these developments, the distance zeta function and its relatives provide a unified analytic mechanism by which tubular geometry, dimension theory, and oscillatory scaling laws are encoded in a meromorphic object. The core principle remains unchanged: geometry is transferred into complex analysis, and singularities of the resulting zeta function reveal the fractal structure.