--- title: 'Tube Zeta Functions: Theory and Applications' url: https://www.emergentmind.com/topics/tube-zeta-functions type: topic --- # Tube Zeta Functions: Theory and Applications 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 [1207.6681]. 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 [1502.00878]. ## 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)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 \(\Re s\) sufficiently large [1502.00878]. The dependence on \(\delta\) is inessential: changing \(\delta\) only alters \(\widetilde\zeta_A\) by an entire function, so the poles and their multiplicities do not depend on \(\delta\) [1502.00878]. The theory extends to relative fractal drums (RFDs), that is, pairs \((A,\Omega)\) with \(A\subset\mathbb{R}^N\), \(\Omega\subset\mathbb{R}^N\) a Borel or measurable set of finite measure, and \(\Omega\subset A_\delta\) for some \(\delta>0\). The relative tube function is \[ V_{A,\Omega}(t):=|A_t\cap\Omega|, \] and the relative tube zeta function is \[ \widetilde\zeta_A(s,\Omega):=\int_0^\delta t^{\,s-N-1}|A_t\cap\Omega|\,dt \] [1502.00878]. In this relative setting, the corresponding box dimension \(\dim_B(A,\Omega)\) may be negative, even \(-\infty\), a phenomenon specific to relative fractal drums [1407.8094]. A further extension replaces a bounded set by the point at infinity. For measurable \(\Omega\subset\mathbb{R}^N\) with \(|\Omega|<\infty\), the tail set is \(\Omega_t:=B_t(0)^c\cap\Omega\), and the tube zeta function at infinity is \[ \widetilde{\zeta}_{\infty,\Omega}(s;T):=\int_T^{+\infty} t^{-s-N-1}\,|\Omega_t|\,dt. \] In this setting, the Minkowski dimensions at infinity are always \(\le -N\) when \(|\Omega|<\infty\) [2208.11245]. ## 2. Relation to distance zeta functions The distance zeta function of a bounded set \(A\subset\mathbb{R}^N\) is \[ \zeta_A(s) := \int_{A_\delta} d(x,A)^{\,s-N}\,dx, \] defined for \(\Re s\) sufficiently large [1502.00878]. Tube and distance zeta functions are linked by the functional equation \[ \zeta_A(s,A_\delta) = \delta^{\,s-N}|A_\delta| + (N-s)\,\widetilde\zeta_A(s,\delta), \] valid for all \(s\) with \(\Re s > \overline{\dim}_B A\) [1502.00878]. In consequence, once one of the two zeta functions is meromorphic on a domain \(U\subset\mathbb{C}\), the identity continues to hold throughout \(U\) by analytic continuation [1502.00878]. This identity implies that tube and distance zeta functions contain essentially the same information. In particular, the abscissae of convergence coincide: \[ D(\widetilde\zeta_A)=D(\zeta_A)=\overline{\dim}_B A, \] and the same holds for relative fractal drums [1502.00878]. Near any pole \(s_0\), a simple pole of one corresponds to a simple pole of the other, differing only by the factor \(N-s\) [1207.6681]. In the bounded case, the poles of \(\zeta_A\) and \(\widetilde\zeta_A\) in a given domain differ only by the trivial factor \((N-s)\), so their visible and principal complex dimensions coincide whenever \(D\overline{\dim}_B A\}\) [1502.00878]. Since \(\overline{\dim}_M A=\overline{\dim}_B A\), the abscissa of convergence of \(\widetilde\zeta_A\) is also the upper Minkowski dimension [1207.6681]. This identification extends to relative fractal drums: \[ D\big(\widetilde\zeta_A(\cdot,\Omega)\big)=\overline{\dim}_B(A,\Omega), \] and remains valid even when the relative box dimension is negative [1407.8094]. At infinity, the analogous statement is \[ D(\widetilde{\zeta}_{\infty,\Omega})=\overline{\dim}_B(\infty,\Omega), \] with the critical dimensions now lying in \((-\infty,-N]\) [2208.11245]. This analytic characterization has several consequences. First, it provides an alternative definition of Minkowski dimension through a half-plane of convergence [1207.6681]. Second, it identifies the critical line \(\{\Re s=D\}\), where \(D=\overline{\dim}_B A\), as the natural location of the principal complex dimensions [1508.04784]. Third, it makes tube zeta functions natural inputs for fractal tube formulas, because the leading asymptotic behavior of \(V_A(t)\) is encoded precisely at the dominant singularities of \(\widetilde\zeta_A\) [1502.00878]. ## 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 \(A\) is bounded, nondegenerate, and \(D=\dim_B A\) exists with \(D0\). Then \(A\) is Minkowski measurable, \(\widetilde\zeta_A\) extends meromorphically at least to \(\{\Re s>D-\alpha\}\), the only pole in that half-plane is \(s=D\), and it is simple with residue \(\mathcal{M}\) [1508.04784]. The nonmeasurable case is governed by oscillatory tube asymptotics. If \[ |A_t| = t^{N-D}\Bigl(G(\log t^{-1}) + O(t^\alpha)\Bigr)\quad\text{as } t\to0^+, \] where \(G\) is a nonconstant periodic function of minimal period \(T>0\), then \(A\) is Minkowski nondegenerate but not Minkowski measurable, with \[ \mathcal{M}_*^D(A)=\min G,\qquad \mathcal{M}^{*D}(A)=\max G \] [1508.04784]. In this regime, \(\widetilde\zeta_A\) extends meromorphically to \(\{\Re s>D-\alpha\}\), and its poles in this half-plane are \[ \left\{ s_k := D + \frac{2\pi i}{T}k : \widehat{G_0}\!\left(\frac{k}{T}\right)\neq 0,\ k\in\mathbb{Z}\right\}, \] all simple, with residues \[ \operatorname{res}(\widetilde\zeta_A,s_k)=\frac{1}{T}\widehat{G_0}\!\left(\frac{k}{T}\right) \] [1508.04784]. In particular, \[ \operatorname{res}(\widetilde\zeta_A,D)=\frac{1}{T}\int_0^T G(\tau)\,d\tau, \] so the residue at \(D\) becomes the average Minkowski content [1207.6681]. The same dichotomy persists at infinity. If \[ |\Omega_t|=t^{N+D}(\gamma+O(t^{-\alpha}))\quad\text{as }t\to+\infty, \] then \((\infty,\Omega)\) is Minkowski measurable at infinity and \[ \operatorname{res}\bigl(\widetilde{\zeta}_{\infty,\Omega},D\bigr)=\gamma \] [2208.11245]. If instead \[ |\Omega_t|=t^{N+D}\bigl(G(\log t)+O(t^{-\alpha})\bigr), \] with \(G\) periodic, then the poles again form a vertical arithmetic progression on the critical line, and the residue at \(D\) is the mean value of \(G\) [2208.11245]. ## 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 [1508.04784]. The principal complex dimensions are those with real part equal to the upper box dimension \(D\) [1502.00878]. Their geometric significance is that each pole \(\omega\) contributes a term of the form \(t^{N-\omega}\) to the tube asymptotics, with nonreal \(\omega\) producing oscillations in \(\log t^{-1}\) [1508.04784]. Under languidity conditions, one obtains explicit fractal tube formulas for relative fractal drums: \[ |A_t\cap\Omega| = \sum_{\omega\in P(\widetilde{\zeta}_A(\cdot,\Omega),W)} \operatorname{res}\left( t^{N-s}\widetilde{\zeta}_A(s,\Omega),\,\omega\right) + \mathcal{R}(t), \] where \(\mathcal{R}(t)\) is an error term controlled by the screen defining the window \(W\) [1502.00878]. If all poles are simple, the contribution of \(\omega\) reduces to \(c_\omega t^{N-\omega}\), where \(c_\omega=\operatorname{res}(\widetilde{\zeta}_A,\omega)\) [1502.00878]. For generalized von Koch snowflakes, scaling functional equations lead to a concrete tube-zeta representation. If \(K_{n,r}\) is a generalized von Koch snowflake with \(\ell=(1-r)/2\), then the relative tube zeta function of \((K_{n,r},\Omega)\) satisfies \[ \tilde\zeta_{K_{n,r},\Omega}(s) = \frac{1}{1-2\ell^s-(n-1)r^s}\,\Big( E(s;\delta) + \tilde\zeta_R(s;\delta) \Big), \] and the possible complex dimensions in \(\Re s>0\) are among the solutions of \[ 1=2\ell^\omega+(n-1)r^\omega \] [2405.04712]. The corresponding tube formula writes the tube function as a sum over residues \[ V_{K_{n,r},\Omega}(t) = \sum_{\omega} a_\omega t^{2-\omega} + R_{K_{n,r},\Omega}(t), \] at least in a distributional sense [2405.04712]. This suggests a general principle: self-similar scaling laws produce Moran-type equations for complex dimensions, while the Mellin-transform nature of \(\widetilde\zeta_A\) converts those scaling laws into meromorphic continuation and residue expansions [2405.04712]. ## 6. Representative examples and developments The classical examples already display the main phenomena. For the Sierpiński carpet \(A\subset\mathbb{R}^2\), one has \[ D=\log_3 8,\qquad T=\log 3, \] and both \(\widetilde\zeta_A\) and \(\zeta_A\) admit meromorphic continuation to all of \(\mathbb{C}\), with principal complex dimensions \[ \left\{ D + \frac{2\pi i}{T}k : k\in\mathbb{Z}\right\} \] [1207.6681]. This is the model lattice case: the critical line supports a vertical arithmetic progression of simple poles. For the middle-third Cantor set \(C\subset\mathbb{R}\), the tube function has log-periodic behavior, \(D=\log_3 2\), and the tube zeta function has the same pole pattern as the distance zeta function: a vertical lattice of simple poles on \(\Re s=D\) [1207.6681]. Generalized Cantor sets \(C(m,a)\) exhibit the same structure, with \[ D=\log_{1/a}m,\qquad T=\log(1/a), \] and poles at \[ D+\frac{2\pi i}{T}\mathbb{Z} \] [1508.04784]. Smooth sets yield the opposite extreme. For the sphere \(A=\partial B_R(0)\subset\mathbb{R}^N\), \(\widetilde\zeta_A\) is meromorphic on all of \(\mathbb{C}\), \(\dim_B A=N-1\), and the poles are real: \[ \{N-1,N-3,\dots\} \] [1407.8094]. 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 [2208.11245]. The latter are sets for which every point on the critical line is a nonremovable singularity of the corresponding fractal zeta function [1502.00878]. A distinct dynamical application concerns orbits of parabolic germs of diffeomorphisms. Their relative tube zeta functions admit meromorphic extension to all of \(\mathbb{C}\), and their poles encode the formal class of the germ [2010.05955]. Notably, these orbits provide examples with nontrivial Minkowski dimension and higher-order oscillatory terms in the tube function, but no nonreal complex dimensions [2010.05955]. 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 [1502.00878]. Source: https://www.emergentmind.com/topics/tube-zeta-functions