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Register a new userDistance and tube zeta functions of fractals and arbitrary compact sets
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Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${mathbb R}^N$, for any integer $Nge1$. It is defined by $zeta_A(s)=int_{A_{delta}}d(x,A)^{s-N},mathrm{d} x$ for all $sinmathbb{C}$ with $operatorname{Re},s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{delta}$ is the $delta$-neighborhood of $A$, where $delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $zeta_A$ is equal to $overlinedim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $zeta_A$ located on the critical line ${mathop{mathrm{Re}} s=overlinedim_BA}$, provided $zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $tildezeta_A(s)=int_0^{delta} t^{s-N-1}|A_t|,mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $tildezeta_A$ computed at $D=dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Bakers theorem from the theory of transcendental numbers.
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We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $zeta_A(s):=int_{A_delta} d(x,A)^{s-N}mathrm{d}x$, where $delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $mathbb{R}^N$. The abscissa of Lebesgue convergence $D(zeta_A)$ coincides with $D:=overlinedim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the critical line ${Re(s)=D}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $tto0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(mathcal M+O(t^gamma))$ as $tto0^+$, with $gamma>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(log t^{-1})+O(t^gamma))$ as $tto0^+$, where $G$ is a nonconstant periodic function and $gamma>0$. In both cases, we show that $zeta_A$ can be meromorphically extended (at least) to the open right half-plane ${Re(s)>D-gamma}$. Furthermore, up to a multiplicative constant, the residue of $zeta_A$ evaluated at $s=D$ is shown to be equal to $mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line ${Re(s)=D}$.
We study fractality of unbounded sets of finite Lebesgue measure at infinity by introducing the notions of Minkowski dimension and content at infinity. We also introduce the Lapidus zeta function at infinity, study its properties and demonstrate its use in analysis of fractal properties of unbounded sets at infinity.
We study the essential singularities of geometric zeta functions $zeta_{mathcal L}$, associated with bounded fractal strings $mathcal L$. For any three prescribed real numbers $D_{infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{infty}<D_1le D$, we construct a bounded fractal string $mathcal L$ such that $D_{rm par}(zeta_{mathcal L})=D_{infty}$, $D_{rm mer}(zeta_{mathcal L})=D_1$ and $D(zeta_{mathcal L})=D$. Here, $D(zeta_{mathcal L})$ is the abscissa of absolute convergence of $zeta_{mathcal L}$, $D_{rm mer}(zeta_{mathcal L})$ is the abscissa of meromorphic continuation of $zeta_{mathcal L}$, while $D_{rm par}(zeta_{mathcal L})$ is the infimum of all positive real numbers $alpha$ such that $zeta_{mathcal L}$ is holomorphic in the open right half-plane ${{rm Re}, s>alpha}$, except for possible isolated singularities in this half-plane. Defining $mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{infty}$ of essential singularities of $zeta_{mathcal L}$, contained in the open right half-plane ${{rm Re}, s>D_{infty}}$, coincides with the vertical line ${{rm Re}, s=D_{infty}}$. We extend this construction to the case of distance zeta functions $zeta_A$ of compact sets $A$ in $mathbb{R}^N$, for any positive integer $N$.
We study a log-gas on a network (a finite, simple graph) confined in a bounded subset of a local field (i.e. R, C, Q_{p} the field of p-adic numbers). In this gas, a log-Coulomb interaction between two charged particles occurs only when the sites of the particles are connected by an edge of the network. The partition functions of such gases turn out to be a particular class of multivariate local zeta functions attached to the network and a positive test function which is determined by the confining potential. The methods and results of the theory of local zeta functions allow us to establish that the partition functions admit meromorphic continuations in the parameter b{eta} (the inverse of the absolute temperature). We give conditions on the charge distributions and the confining potential such that the meromorphic continuations of the partition functions have a pole at a positive value b{eta}_{UV}, which implies the existence of phase transitions at finite temperature. In the case of p-adic fields the meromorphic continuations of the partition functions are rational functions in the variable p^{-b{eta}}. We give an algorithm for computing such rational functions. For this reason, we can consider the p-adic log-Coulomb gases as exact solvable models. We expect that all these models for different local fields share common properties, and that they can be described by a uniform theory.
We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,Omega)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,Omega)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $Omega$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,Omega)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpinski gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devils staircase) RFD, which is shown to be `subcritically fractal.
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