Tube Zeta Functions: Theory and Applications

• Tube zeta functions are Mellin-type transforms that encode the small-scale behavior of tubular neighborhoods and recover the Minkowski dimension as the abscissa of convergence.
• The approach uses analytic continuation to relate tube and distance zeta functions, unifying geometric invariants and complex dimensions for various fractal sets.
• Residues at critical poles quantify the Minkowski content, leading to explicit fractal tube formulas and insights into oscillatory scaling behaviors.

Searching arXiv for recent and foundational papers on tube zeta functions. Tube zeta functions are Mellin-type transforms of tube volumes attached to bounded subsets of Euclidean space and, more generally, to relative fractal drums. In the Lapidus–Radunović–Žubrinić framework, they encode the small-scale behavior of tubular neighborhoods, recover the upper box or Minkowski dimension as an abscissa of convergence, and relate residues at critical poles to Minkowski content (Lapidus et al., 2012). Closely related distance zeta functions carry essentially the same meromorphic data through an explicit functional equation, so tube zeta functions serve both as intrinsic geometric invariants and as analytic vehicles for defining complex dimensions of fractal sets (Lapidus et al., 2015).

1. Definition and ambient setting

For a bounded set $A\subset \mathbb{R}^N$, its open $t$-neighborhood is

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

where $d(x,A)$ denotes the Euclidean distance from $x$ to $A$. The associated tube function is the Lebesgue measure $V_A(t):=|A_t|$ (Lapidus et al., 2015). With $\delta>0$ fixed, the tube zeta function is defined by

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

for all $s\in\mathbb{C}$ with $t$0 sufficiently large (Lapidus et al., 2015). The dependence on $t$1 is inessential: changing $t$2 only alters $t$3 by an entire function, so the poles and their multiplicities do not depend on $t$4 (Lapidus et al., 2015).

The theory extends to relative fractal drums (RFDs), that is, pairs $t$5 with $t$6, $t$7 a Borel or measurable set of finite measure, and $t$8 for some $t$9. The relative tube function is

$A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$0

and the relative tube zeta function is

$A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$1

(Lapidus et al., 2015). In this relative setting, the corresponding box dimension $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$2 may be negative, even $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$3, a phenomenon specific to relative fractal drums (Lapidus et al., 2014).

A further extension replaces a bounded set by the point at infinity. For measurable $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$4 with $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$5, the tail set is $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$6, and the tube zeta function at infinity is

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

In this setting, the Minkowski dimensions at infinity are always $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$8 when $A_t:=\{x\in\mathbb{R}^N : d(x,A)<t\},$9 (Radunović, 2022).

2. Relation to distance zeta functions

The distance zeta function of a bounded set $d(x,A)$0 is

$d(x,A)$1

defined for $d(x,A)$2 sufficiently large (Lapidus et al., 2015). Tube and distance zeta functions are linked by the functional equation

$d(x,A)$3

valid for all $d(x,A)$4 with $d(x,A)$5 (Lapidus et al., 2015). In consequence, once one of the two zeta functions is meromorphic on a domain $d(x,A)$6, the identity continues to hold throughout $d(x,A)$7 by analytic continuation (Lapidus et al., 2015).

This identity implies that tube and distance zeta functions contain essentially the same information. In particular, the abscissae of convergence coincide: $d(x,A)$8 and the same holds for relative fractal drums (Lapidus et al., 2015). Near any pole $d(x,A)$9, a simple pole of one corresponds to a simple pole of the other, differing only by the factor $x$0 (Lapidus et al., 2012). In the bounded case, the poles of $x$1 and $x$2 in a given domain differ only by the trivial factor $x$3, so their visible and principal complex dimensions coincide whenever $x$4 (Lapidus et al., 2015).

At infinity the same pattern persists. For $x$5,

$x$6

so one can again pass between distance and tube zeta functions, and the same complex dimensions appear for both (Radunović, 2022).

3. Abscissa of convergence and box or Minkowski dimension

A central theorem of the theory is that the abscissa of convergence of the tube zeta function is geometric rather than merely analytic. For a bounded set $x$7,

$x$8

and $x$9 is holomorphic on $A$0 (Lapidus et al., 2015). Since $A$1, the abscissa of convergence of $A$2 is also the upper Minkowski dimension (Lapidus et al., 2012).

This identification extends to relative fractal drums: $A$3 and remains valid even when the relative box dimension is negative (Lapidus et al., 2014). At infinity, the analogous statement is

$A$4

with the critical dimensions now lying in $A$5 (Radunović, 2022).

This analytic characterization has several consequences. First, it provides an alternative definition of Minkowski dimension through a half-plane of convergence (Lapidus et al., 2012). Second, it identifies the critical line $A$6, where $A$7, as the natural location of the principal complex dimensions (Lapidus et al., 2015). Third, it makes tube zeta functions natural inputs for fractal tube formulas, because the leading asymptotic behavior of $A$8 is encoded precisely at the dominant singularities of $A$9 (Lapidus et al., 2015).

4. Residues, Minkowski content, and the measurable/nonmeasurable dichotomy

For Minkowski measurable sets, the residue of the tube zeta function at the critical dimension recovers Minkowski content exactly. If $V_A(t):=|A_t|$0 is bounded, nondegenerate, and $V_A(t):=|A_t|$1 exists with $V_A(t):=|A_t|$2, then

$V_A(t):=|A_t|$3

and if $V_A(t):=|A_t|$4 is Minkowski measurable,

$V_A(t):=|A_t|$5

(Lapidus et al., 2015). The corresponding result for the distance zeta function differs by the factor $V_A(t):=|A_t|$6 (Lapidus et al., 2012).

A sufficient asymptotic condition for this situation is

$V_A(t):=|A_t|$7

with $V_A(t):=|A_t|$8. Then $V_A(t):=|A_t|$9 is Minkowski measurable, $\delta>0$0 extends meromorphically at least to $\delta>0$1, the only pole in that half-plane is $\delta>0$2, and it is simple with residue $\delta>0$3 (Lapidus et al., 2015).

The nonmeasurable case is governed by oscillatory tube asymptotics. If

$\delta>0$4

where $\delta>0$5 is a nonconstant periodic function of minimal period $\delta>0$6, then $\delta>0$7 is Minkowski nondegenerate but not Minkowski measurable, with

$\delta>0$8

(Lapidus et al., 2015). In this regime, $\delta>0$9 extends meromorphically to $$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$0, and its poles in this half-plane are

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

all simple, with residues

$$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$2

(Lapidus et al., 2015). In particular,

$$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$3

so the residue at $$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$4 becomes the average Minkowski content (Lapidus et al., 2012).

The same dichotomy persists at infinity. If

$$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$5

then $$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$6 is Minkowski measurable at infinity and

$$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$7

(Radunović, 2022). If instead

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

with $$\widetilde\zeta_A(s) := \int_0^\delta t^{\,s-N-1} |A_t|\,dt,$$9 periodic, then the poles again form a vertical arithmetic progression on the critical line, and the residue at $s\in\mathbb{C}$0 is the mean value of $s\in\mathbb{C}$1 (Radunović, 2022).

5. Complex dimensions and fractal tube formulas

Complex dimensions are the poles of a meromorphic extension of a tube or distance zeta function to a suitable neighborhood of the critical line (Lapidus et al., 2015). The principal complex dimensions are those with real part equal to the upper box dimension $s\in\mathbb{C}$2 (Lapidus et al., 2015). Their geometric significance is that each pole $s\in\mathbb{C}$3 contributes a term of the form $s\in\mathbb{C}$4 to the tube asymptotics, with nonreal $s\in\mathbb{C}$5 producing oscillations in $s\in\mathbb{C}$6 (Lapidus et al., 2015).

Under languidity conditions, one obtains explicit fractal tube formulas for relative fractal drums: $s\in\mathbb{C}$7 where $s\in\mathbb{C}$8 is an error term controlled by the screen defining the window $s\in\mathbb{C}$9 (Lapidus et al., 2015). If all poles are simple, the contribution of $t$00 reduces to $t$01, where $t$02 (Lapidus et al., 2015).

For generalized von Koch snowflakes, scaling functional equations lead to a concrete tube-zeta representation. If $t$03 is a generalized von Koch snowflake with $t$04, then the relative tube zeta function of $t$05 satisfies

$t$06

and the possible complex dimensions in $t$07 are among the solutions of

$t$08

(Hoffer, 2024). The corresponding tube formula writes the tube function as a sum over residues

$t$09

at least in a distributional sense (Hoffer, 2024).

This suggests a general principle: self-similar scaling laws produce Moran-type equations for complex dimensions, while the Mellin-transform nature of $t$10 converts those scaling laws into meromorphic continuation and residue expansions (Hoffer, 2024).

6. Representative examples and developments

The classical examples already display the main phenomena. For the Sierpiński carpet $t$11, one has

$t$12

and both $t$13 and $t$14 admit meromorphic continuation to all of $t$15, with principal complex dimensions

$t$16

(Lapidus et al., 2012). This is the model lattice case: the critical line supports a vertical arithmetic progression of simple poles.

For the middle-third Cantor set $t$17, the tube function has log-periodic behavior, $t$18, and the tube zeta function has the same pole pattern as the distance zeta function: a vertical lattice of simple poles on $t$19 (Lapidus et al., 2012). Generalized Cantor sets $t$20 exhibit the same structure, with

$t$21

and poles at

$t$22

(Lapidus et al., 2015).

Smooth sets yield the opposite extreme. For the sphere $t$23, $t$24 is meromorphic on all of $t$25, $t$26, and the poles are real: $t$27 (Lapidus et al., 2014). No oscillatory complex dimensions occur, reflecting the absence of self-similar log-periodicity.

The relative theory adds further flexibility. Relative fractal drums permit negative box dimensions and exact scaling relations, while the theory at infinity produces quasiperiodic and maximally hyperfractal examples whose critical line is a natural boundary (Radunović, 2022). The latter are sets for which every point on the critical line is a nonremovable singularity of the corresponding fractal zeta function (Lapidus et al., 2015).

A distinct dynamical application concerns orbits of parabolic germs of diffeomorphisms. Their relative tube zeta functions admit meromorphic extension to all of $t$28, and their poles encode the formal class of the germ (Mardešić et al., 2020). Notably, these orbits provide examples with nontrivial Minkowski dimension and higher-order oscillatory terms in the tube function, but no nonreal complex dimensions (Mardešić et al., 2020). This shows that oscillatory terms in tube asymptotics need not always correspond to nonreal poles, a useful qualification to the usual self-similar paradigm.

Across these examples, tube zeta functions unify several themes: analytic characterization of dimension, residue formulas for Minkowski content, explicit fractal tube formulas, and a pole calculus that detects both periodic and quasiperiodic geometric oscillations (Lapidus et al., 2015).