Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality

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 Sierpi\'nski 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 devil's staircase) RFD, which is shown to be `subcritically fractal'.Comment: 90 pages (because of different style file), 5 figures, corrected typos, updated reference

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

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 $D:=\overline\dim_BA$, 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 $t\to0^+$, 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 $t\to0^+$, 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 $t\to0^+$, 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 $\{\Re(s)=D\}$.Comment: 30 pages, 2 figures, improved parts of the paper and shortened the paper by reducing background material, to appear in Journal of mathematical analysis and applications in 201

Essential singularities of fractal zeta functions

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 $D_{\infty}<D_1\le D$, 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 $N$.Comment: Theorem 3.2 (b) was wrong in the previous version, so we have decided to omit it and pursue this issue at some future time. Part (b) of Theorem 3.2. was not used anywhere else in the paper. Theorem 3.2. is now called Proposition 3.2. on page 12. Corrected minor typos and added new references To appear in: Pure and Applied Functional Analysis; issue 5 of volume 5 (2020

SINGULAR DIMENSION OF SPACES OF REAL FUNCTIONS

Let X be a space of measurable real functions defined on a fixed open set Ω ⊆ R^N . It is natural to define the singular dimension of X as the supremum of Hausdorff dimension of singular sets of all functions in X.We say that f ∈ X is a maximally singular function in X if the Hausdorff dimension of its singular set is the largest possible. The paper discusses recent results about singular dimension of Banach spaces of functions, existence and density of maximally singular functions, and provides some open problems.Let X be a space of measurable real functions defined on a fixed openset Ω&nbsp;C RN. It is natural to define the singular dimension of X as thesupremum of Hausdorff dimension of singular sets of all functions in X.We say that f&nbsp;C X is a maximally singular function in X if the Hausdorffdimension of its singular set is the largest possible. The paper discusses recent results about singular dimension of Banach spaces of functions, existence and density of maximally singular functions, and provides some open problems