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We give a generalization of Lagarias' formula for diffraction by ideal crystals, and we apply it to the lattice case, in preparation for addressing the problem of quasicrystals and complex dimensions posed by Lapidus and van Frankenhuijsen concerning the quasiperiodic properties of the set of complex dimensions of any nonlattice self-similar fractal string. More specifically, in this paper, we consider the case of the complex dimensions of a lattice (rather than of a nonlattice) self-similar string and show that the corresponding diffraction measure exists, is unique, and is given by a suitable $\textit{continuous}$ analogue of a discrete Dirac comb. We also obtain more general results concerning the autocorrelation measures and diffraction measures of generalized idealized fractals associated to possibly degenerate lattices and the corresponding extension of the Poisson Summation Formula., Comment: A new version with a few updates, 27 pages, 4 figures. arXiv admin note: text overlap with arXiv:2009.03493

The local theory of complex dimensions for real and $p$-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for ad\`elic fractal strings in order to reveal the oscillatory nature of ad\`elic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar $p$-adic fractal strings of any rational dimension in the closed unit interval $[0,1]$. Moreover, as a first step towards a global theory of complex dimensions for ad\`elic fractal strings, we construct an ad\`elic Cantor string in the set of finite ad\`eles $\mathbb{A}_0$ as an infinite Cartesian product of every $p$-adic Cantor string, as well as an ad\`elic Cantor-Smith string in the ring of ad\`eles $\mathbb{A}$ as a Cartesian product of the general Cantor string and the ad\`elic Cantor string., Comment: 16 pages

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpi\'nski is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications., Comment: 35 pages, 3 figures. To appear in Adv. Math

The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lov\'asz., Comment: 38 pages, 11 figures, 7 tables

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 $D_{\infty}\alpha\}$, 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)

Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), in \S2, and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of $\mbr^N$, for any integer $N \geq 1$, in \S3. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs, for which we refer to the author's previous (joint) books mentioned in the paper. Finally, in an epilogue (\S4), entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case) [HerLap1], along with [Lap8], and with T. Cobler (in the complex case) [CobLap1], respectively, as well as in the latter part of a book in preparation by the author, [Lap10]., Comment: To appear in: {\em Horizons of Fractal Geometry and Complex Dimensions} (R.~G. Niemeyer, E.~P.~J. Pearse, J.~A. Rock and T. Samuel, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, R.~I., 2019

Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil for curves and later, by Deligne for varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to those of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis is still on. In this paper, we study the properties of the derivative operator $D = \frac{d}{dz}$ on a particular weighted Bergman space of entire functions. The operator $D$ can be naturally viewed as the `infinitesimal shift of the complex plane'. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated to any suitable meromorphic function. We then show that the meromorphic function can be recovered by using a regularized determinant involving the above operator. This is illustrated in some important special cases: rational functions, zeta functions of curves over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. Our construction is motivated in part by [23] on the infinitesimal shift of the real line, as well as by earlier work of Deninger [10] on cohomology in number theory and a conjectural `fractal cohomology theory' envisioned in [25] and [28].

We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measurability criterion established here is a generalization of the corresponding one obtained for fractal strings by the first author and M.\ van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion established here is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function. We also reflect on the notion of gauge-Minkowski measurability of RFDs and establish several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples., Comment: 53 pages, modified and shortened exposition, corrected typos. arXiv admin note: text overlap with arXiv:1604.08014

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 references

The local theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a $p$-adic fractal string $\mathcal{L}_p$, expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a $p$-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and $p$-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case., Comment: 34 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1105.2966 This is the final version of an original research article on the Minkowski dimension and explicit tube formulas for $p$-adic fractal strings. It is appeared in the open access journal Fractal Fractional

In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions', associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions' of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums' (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $\infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {\em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $m\ge1$, and even an infinite sequence of essential singularities along the critical line., Comment: 101 pages, corrected typos

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 2017

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 $N\ge1$. It is defined by $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\,\mathrm{d} x$ for all $s\in\mathbb{C}$ 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 $\overline\dim_BA$, 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 $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tilde\zeta_A(s)=\int_0^{\delta} t^{s-N-1}|A_t|\,\mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tilde\zeta_A$ 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 Baker's theorem from the theory of transcendental numbers., Comment: 54 pages, corrected misprints, reduced number of self-citations

We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments, including the theory of (fractal) complex dimensions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his collaborators)., Comment: 55 pages, to appear in Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday (C. B. Pomerance and M. T. Rassias, eds.)

We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we examine the limiting behavior of particular sequences of compatible periodic orbits and, more interesting, in Section 5, the limiting behavior of a particular sequence of compatible singular orbits. The latter seems to indicate that the classification of orbits may not be so straightforward. Additionally, sufficient conditions for the existence of particular nontrivial paths is given in Section 4. The proofs of two results stated in [LapNie4] appear here for the first time, as well. A discussion of our results and directions for future research is then given in Section 6., Comment: 20 Figures, 35 pages, two results from arXiv:1210.0282 are generalized and proved in this article. Many new results appear here. Comments welcome. Version 3 contains minor grammatical changes and the presentation of some results has greatly improved. To appear in the journal Nonlinearity

In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space)., Comment: For inclusion in the Proceedings of the International Conference "Geometry and Stochastics V," Tabarz, Germany, March 2014, Progress in Probability, Birkh\"auser, Basel, Boston and Berlin, 2015, C. Bandt, K. Falconer and M. Z\"ahle, eds.; based on a plenary lecture given by the first author at that conference, 29 pages, corrected Equations (26) and (27) and the 2nd sentence after Equation (46)

This research expository article contains a survey of earlier work (in \S2--\S4) but also contains a main new result (in \S5), which we first describe. Given $c \geq 0$, the spectral operator $\mathfrak{a} = \mathfrak{a}_c$ can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding $c$. Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function $\zeta = \zeta(s)$: $\mathfrak{a} = \zeta (\partial)$, where $\partial = \partial_c$ is the infinitesimal shift of the real line acting on the weighted Hilbert space $L^2 (\mathbb{R}, e^{-2ct} dt)$. In this paper, we establish a new asymmetric criterion for the Riemann hypothesis, expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter $c \in (0, 1/2)$ (i.e., for all $c$ in the left half of the critical interval $(0,1)$). This corresponds (conditionally) to a mathematical (and perhaps also, physical) "phase transition" occurring in the midfractal case when $c= 1/2$. Both the universality and the non-universality of $\zeta = \zeta (s)$ in the right (resp., left) critical strip $\{1/2 < \text{Re}(s) < 1 \}$ (resp., $\{0 < \text{Re}(s) < 1/2 \}$) play a key role in this context. These new results are presented in \S5. In \S2, we briefly discuss earlier joint work on the complex dimensions of fractal strings, while in \S3 and \S4, we survey earlier related work of the author with H. Maier and with H. Herichi, respectively, in which were established symmetric criteria for the Riemann hypothesis, expressed respectively in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension $D \in (0,1),$ with $D \neq 1/2$, and of the quasi-invertibility of the family of spectral operators $\mathfrak{a}_c$ (with $c \in (0,1), c \neq 1/2$)., Comment: 26 pages. To appear in the special issue of the "Philosophical Transactions of the Royal Society", Ser. A., titled "Geometric Concepts in the Foundations of Physics", 2015

We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion., Comment: 15 pages, corrected typos, updated references, accepted for publication in Discrete and Continuous Dynamical Systems - Series S

The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area., Comment: 60 pages. Added references. Corrected typos. Expanded Thm. 2.9 to the special case when D=N, see also Rem. 2.10. Added Rem. 2.6, 3.2, 3.9, 5.17 and Rem. 8.9 corresponding to Prob. 8.8. Slightly updated Defs. 5.1, 5.20 and expanded Exercise 5.22. The paper is to appear in the Festschrift volume of the Journal of Fixed Point Theory and Applications in honor of Haim Brezis' 70th birthday

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an essential use of the functional analytic framework developed by the authors in \cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial$ of the real line: $\mathfrak{a}=\zeta(\partial)$. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta(s)$ proposed here, the role played by the complex variable $s$ in the classical universality theorem is now played by the family of `truncated infinitesimal shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral operator in connection with a spectral reformulation of the Riemann hypothesis as an inverse spectral problem for fractal strings. This latter work provided an operator-theoretic version of the spectral reformulation obtained by the second author and H. Maier in \cite{LaMa2}. In the long term, our work (along with \cite{La5, La6}), is aimed in part at providing a natural quantization of various aspects of analytic number theory and arithmetic geometry.

In this survey article, we investigate the spectral properties of fractal differential operators on self-similar fractals. In particular, we discuss the decimation method, which introduces a renormalization map whose dynamics describes the spectrum of the operator. In the case of the bounded Sierpinski gasket, the renormalization map is a polynomial of one variable on the complex plane. The decimation method has been generalized by C. Sabot to other fractals with blow-ups and the resulting associated renormalization map is then a multi-variable rational function on a complex projective space. Furthermore, the dynamics associated with the iteration of the renormalization map plays a key role in obtaining a suitable factorization of the spectral zeta function of fractal differential operators. In this context, we discuss the works of A. Teplyaev and of the authors regarding the examples of the bounded and unbounded Sierpinski gaskets as well as of fractal Sturm-Liouville differential operators on the half-line., Comment: 23 pages, 4 figures

We construct Dirac operators and spectral triples for certain, not necessarily self-similar, fractal sets built on curves. Connes' distance formula of noncommutative geometry provides a natural metric on the fractal. To motivate the construction, we address Kigami's measurable Riemannian geometry, which is a metric realization of the Sierpinski gasket as a self-affine space with continuously differentiable geodesics. As a fractal analog of Connes' theorem for a compact Riemmanian manifold, it is proved that the natural metric coincides with Kigami's geodesic metric. This present work extends to the harmonic gasket and other fractals built on curves a significant part of the earlier results of E. Christensen, C. Ivan, and the first author obtained, in particular, for the Euclidean Sierpinski gasket. (As is now well known, the harmonic gasket, unlike the Euclidean gasket, is ideally suited to analysis on fractals. It can be viewed as the Euclidean gasket in harmonic coordinates.) Our current, broader framework allows for a variety of potential applications to geometric analysis on fractal manifolds., Comment: 39 pages, 2 Figures. We have changed the margin size, numbered definitions, lemmas, remarks, and corollaries within sections, updated a reference, and corrected a few typos from the second version

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. This problem is related to the question "Can one hear the shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer for fractal strings whose dimension is $c\in(0,1)-\{1/2}$ if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of a fractal string onto its spectrum. We focus here on presenting the rigorous results obtained by the authors in [HerLa1] about the invertibility of the spectral operator. We show that given any $c\geq0$, the spectral operator $\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normal operator acting in a Hilbert space $\mathbb{H}_{c}$, is `quasi-invertible' (i.e., its truncations are invertible) if and only if the Riemann zeta function $\zeta=\zeta(s)$ does not have any zeroes on the line $Re(s)=c$. It follows that the associated inverse spectral problem has a positive answer for all possible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$, if and only if the Riemann hypothesis is true., Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M. van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 2013. arXiv admin note: substantial text overlap with arXiv:1203.4828

If D is a rational polygon, then the associated rational billiard table is given by \Omega(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding fractal billiard table is denoted by \Omega(F). In this paper, we survey many of the results from [LapNie1-3] for the Koch snowflake fractal billiard \Omega(KS) and announce new results on two other fractal billiard tables, namely, the T-fractal billiard table \Omega(T) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table \Omega(S_a) (see [CheNie]). We build a general framework within which to analyze what we call a sequence of compatible orbits. Properties of particular sequences of compatible orbits are discussed for each prefractal billiard \Omega(KS_n), \Omega(T_n) and \Omega(S_a,n), for n = 0, 1, 2... . In each case, we are able to determine a particular limiting behavior for an appropriately formulated sequence of compatible orbits. Such a limit either constitutes what we call a nontrivial path of a fractal billiard table \Omega(F) or else a periodic orbit of \Omega(F) with finite period. In our examples, F will be either KS, T or S_a. Several of the results and examples discussed in this paper are presented for the first time. We then close with a brief discussion of open problems and directions for further research in the emerging field of fractal billiards., Comment: 26 figures. his article is a survey of resent results, in addition to newly announced results, proofs for which are to appear in forthcoming papers; arXiv admin note: text overlap with arXiv:1204.3133

The theory of p-adic fractal strings and their complex dimensions was developed by the first two authors in [17, 18, 19], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [28]. Using the fractal tube formula obtained by the authors for p-adic fractal strings in [20], we present here an exact volume formula for the tubular neighborhood of a p-adic self-similar fractal string Lp, expressed in terms of the underlying complex dimensions. The periodic structure of the complex dimensions allows one to obtain a very concrete form for the resulting fractal tube formula. Moreover, we derive and use a truncated version of this fractal tube formula in order to show that Lp is not Minkowski measurable and obtain an explicit expression for its average Minkowski content. The general theory is illustrated by two simple examples, the 3-adic Cantor string and the 2-adic Fibonacci strings, which are nonarchimedean analogs (introduced in [17, 18]) of the real Cantor and Fibonacci strings studied in [28]., Comment: 24 pages, 4 figures, submitted to Contemporary Mathematics by the AMS. arXiv admin note: substantial text overlap with arXiv:1105.2966

We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions; the similarity dimension via the Moran equation (at least in the case of self-similar sets); the order of the (box-)counting function; the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings; and the abscissae of convergence of new classes of zeta functions. Specifically, we define box-counting zeta functions of infinite bounded subsets of Euclidean space and discuss results pertaining to distance and tube zeta functions. Appealing to an analysis of these zeta functions allows for the development of theories of complex dimensions for bounded sets in Euclidean space, extending techniques and results regarding (ordinary) fractal strings obtained by the first author and van Frankenhuijsen., Comment: 33 pages, 1 figure

The multifractal structure underlying a self-similar measure stems directly from the weighted self-similar system (or weighted iterated function system) which is used to construct the measure. This follows much in the way that the dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity dimension, is determined by the scaling ratios of the corresponding self-similar system via Moran's theorem. The multifractal structure allows for our definition of scaling regularity and scaling zeta functions motivated by geometric zeta functions and, in particular, partition zeta functions. Some of the results of this paper consolidate and partially extend the results regarding a multifractal analysis for certain self-similar measures supported on compact subsets of a Euclidean space based on partition zeta functions. Specifically, scaling zeta functions generalize partition zeta functions when the choice of the family of partitions is given by the natural family of partitions determined by the self-similar system in question. Moreover, in certain cases, self-similar measures can be shown to exhibit lattice or nonlattice structure with respect to specified scaling regularity values. Additionally, in the context provided by generalized fractal strings viewed as measures, we define generalized self-similar strings, allowing for the examination of many of the results presented here in a specific overarching context and for a connection to the results regarding the corresponding complex dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice strings and recursive strings are defined and shown to be very closely related., Comment: 33 pages, no figures, in press

The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows., Comment: 21 pages, 15 figures

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we also give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function zeta(s) does not have any zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.