Curved model sets and crystalline measures
\textbf{Aim of the paper.} This paper constructs crystalline measures as curved versions of model sets. It relates diverse concepts in quasicrystals, crystallinn measures, and Fourier quasicrystals using a class of measures on tori pioneered by \textit{P. R. Ahern} [Mich. Math. J. 20, 33--37 (1973; Zbl 0265.32003)]. The author concludes Section 1 with the sentence ``Our approach does not yield new results but provides us with a new perspective on the achievements by Kurasov and Sarnak.''\N\N\textbf{Historical Background.} We give a brief historical background emphasizing the seminal contributions by Yves Meyer. Definitions are given in Section 3. In 1959 \textit{A. P. Guinand} [Acta Math. 101, 235--271 (1959; Zbl 0085.30102)] gave an ingenious construction of a crystalline measure on \(\mathbb R\). His derivation had an error that was corrected by \textit{Y. F. Meyer} [Proc. Natl. Acad. Sci. USA 113, No. 12, 3152--3158 (2016; Zbl 1367.28002); Rev. Mat. Iberoam. 33, No. 3, 1025--1036 (2017; Zbl 1384.42009)]. His measure is a Bohr almost periodic distributiion (see Lemma 2) but is not a Bohr almost measure because it is not translation bounded [\textit{Y. F. Meyer}, Proc. Natl. Acad. Sci. USA 113, No. 12, 3152--3158 (2016; Zbl 1367.28002); Bull. Hell. Math. Soc. 61, 11--20 (2017; Zbl 1425.42008)] (see page 394 lines 6--7). It is clearly a Fourier quasicrystal. We note that it is not a positve measure and that all positive Fourier quasicrystals are totally bounded.\N\NAperiodic planar tilings such as those of \textit{R. Penrose} [The role of aesthetics in pure and applied mathematical research'', Bull. Inst. Math. Appl. 10, 266--271 (1974)] and physical quasicrystals synthesized by \textit{D. Schechtman} et al. [Phys. Rev. Lett. 53, No. 1, 1951--1954 (1984; \url{doi:10.1103/PhysRevLett.53.1951})] are described by Meyer's model sets as explained by \textit{J. C. Lagarias} [Commun. Math. Phys. 179, No. 2, 365--376 (1996; Zbl 0858.52010)] and \textit{R. V. Moody} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 489, 403--441 (1997; Zbl 0880.43008)]. \textit{J. C. Lagarias} proved [CRM Monogr. Ser. 13, 61--93 (2000; Zbl 1161.52312)] the measures associated to these model sets are Besicovitch but nor Bohr almost peridodic (see page 400 lines 10--11). Furthermore if \(\Lambda\) is a model set then the Fourier transform \(\widehat \mu\) of \(\mu := \sum_{\lambda \in \Lambda} \delta_\lambda\) is not a measure. This is due to the fact that \(\Lambda\) is constructed from a subset of a torus having a boundaty (see page 400 lines 25--26).\N\NThe simplest examples of a crystalline measure on \(\mathbb R\) is the Dirac comb \(\mu_{DC} := \sum_{k \in \mathbb Z} \delta_k\), since Poisson's formula implies \(\widehat {\mu_{DC}} = \mu_{DC}\), and trivial measures constructed from it. \textit{N. Lev} and \textit{A. Olevskii} [Rev. Mat. Iberoam. 32, No. 4, 1341--1352 (2016; Zbl 1366.42011)] constructed nontrivial crystalline measures \(\mu\) on \(\mathbb R\) and proved that they cannot be supported on a model set. \textit{S. Yu. Favorov} [Anal. Math. 50, No. 2, 455--462 (2024; Zbl 07915684)] constructed a crystalline measure that is not a Fourier quasicrystal \textit{P. Kurasov} and \textit{P. Sarnak} [J. Math. Phys. 61, No. 8, 083501, 13 p. (2020; Zbl 1459.05177)] consructed the first Fourier quasicrystals on \(\mathbb R\) with positive integer coefficients. Their measures are divisors of real-rooted trigonometric polynomials. \textit{A. Olevskii} and \textit{A. Ulanovskii} [C. R., Math., Acad. Sci. Paris 358, No. 11--12, 1207--1211 (2020; Zbl 1456.42032)] proved that all Fourier quasicrystals with positive integer coefficients have this form. These measure are constructed from curved surfaces without boundaries in tori wherease the modelf sets are constructed from flat surfaces in tori with boundaries whence the papers title.\N\N\textbf{Definitions and Basic Facts.} We define main concepts used in the paper and basic facts stated in Sections 1 and 2.\N\NLet \(C_b(\mathbb R)\) be the space of bounded complex-valued continuous functions on \(\mathbb R\) and \(C_c(\mathbb R)\) be its subspace of compactly supported functions. A Radon measure on \(\mathbb R\) is a linear functional \(\mu\) on \(C_c(\mathbb R)\) such that for every \(g \in C_c(\mathbb R)\), \(g\mu\) is a (bounded) Borel measure on \(\mathbb R\). A Radon measure \(\mu\) is translation bounded if its variation measure \(|\mu|\) satisfies \(\sup_{x \in \mathbb R} |\mu|([x,x+1]) < \infty\). This foundational concept was first introduced in 1974 by \textit{L. Argabright} and \textit{J. Gil de Lamadrid} [Fourier analysis of unbounded measures on locally compact Abelian groups. Providence, RI: American Mathematical Society (AMS) (1974; Zbl 0294.43002), page 28]).\N\NIf \(\mu\) is a Radon measure then the convolution \(\mu*f \in C(\mathbb R^n)\) for every \(f \in C_c(\mathbb R^n)\). A Besicovitch, resp. Bohr almost periodic measure is a Radon measure \(\mu\) such that \(\mu*f\) is a Besicovitch, resp. Bohr almost periodic function for every \(f \in C_c(\mathbb R^n)\). Let \(\mathcal S(\mathbb R^n)\) be the space of Schwartz functions and \(\mathcal S^{\prime}(\mathbb R^n)\) it dual space of tempered distributions.\N\NA Besicovitch, resp. Bohr almost periodic distribution is a tempered distribution \(\mu\) such that \(\mu*f\) is a Besicovitch, resp. Bohr almost periodic function for every \(f \in \mathcal S(\mathbb R^n)\). The Banach-Steinhaus theorem implies that a Radon measure is a Bohr almost periodic measure iff it is a translation bounded and a Bohr almost periodic distribution.\N\NA crystalline measures on \(\mathbb R^n\) is a tempered Radon meaure \(\mu\) supported on a discrete set whose Fourier transform \(\widehat \mu\) is also a Radon measure supported on a discrete set. Thus\N\[\N \mu = \sum_{\lambda \in \Lambda} a(\lambda) \delta_\lambda ; \ \ \widehat \mu = \sum_{s \in S} b(s) \delta_s.\tag{1}\N\]\N\(\Lambda\), resp \(S\) is the support, resp. spectrum of \(\mu\). \(\Lambda \subset \mathbb R^n\) is uniformly discrete if\N\[\N\inf \{|x - y| : x, y \in \Lambda, x \neq y\} > 0.\N\]\NLemma 1 shows that if \(\mu\) is a crystalline measure, then both series in (1) converge in \(\mathcal S'(\mathbb R)\). Lemma 2 shows that if \(\mu\) is a positive crystalline measure and \(a(\lambda) \geq 1\), then \(\Lambda\) is a finite union of uniformly discrete sets and \(\mu\) is translation bounded. Lemma 3 show that a translation bounded crystalline measure is a Bohr almost periodic meaure. Lemma 4 defines the Fourier coefficients \(\widehat \mu(\omega)\) of a Bohr almost periodic distribution \(\mu\) by\N\[\N\widehat \mu(\omega) := \widehat {\mu*f}(\omega)\, / \, \widehat f (\omega)\N\]\Nwhere \(f \in \mathcal S(\mathbb R^n)\) and the Fourier transform \(\widehat f\) of \(f\) is nonzero at \(\omega \in \mathbb R\). The reader should note that \(\widehat \mu\) is not the distributional Fourier transform of \(\mu\) and in the literature is sometimes denoted by a different notation and called the Fourier-Bohr transform of \(\mu\). Lemma 4 proves that this definition is independent of the choice of \(g\).\N\NProposition 1 shows that every Bohr almost periodic measure \(\mu\) induces a continuous linear function on the space \(\mathcal {AP}\) of Bohr almost periodic functions on \(\mathbb R\). Since \(\mathcal {AP}\) can be identified with the space of continuous functions on the Bohr compactification \(B(\mathbb R)\) of \(\mathbb R\), the Riesz representation theorem implies that the \(\mu\) defines a measure \(\mu \circ b\) where \(b : \mathbb R \mapsto B(\mathbb R)\) is the canonical continuous homomorphism with dense image and \(\mu \circ b\) is the image of \(\mu\) under the Bohr mapping ([\textit{L. N. Argabright} and \textit{J. G. de Lamadrid}, lmost periodic measures. Providence, Rhode Island (1990)], Chapter 5). Theorem 1 shows that if \(S \subset \mathbb R\) is discrete and \(\mu\) is a translation bounded measure supported on \(S\) then the following two properties are equivalent\N\begin{itemize}\N\item[1.] \(\mu = \sum_{s \in S} b(s)\, \delta_s\) where convergence to \(\mu\) is in \(\mathcal S^{\, \prime}(\mathbb R)\).\N\item[2.] \(\mu\) is a Bohr almost periodic measure and \(\widehat \mu(s) = b(s)\).\N\end{itemize}\N\NLet \(\Gamma\) be the subgroup of \(\mathbb R\) generated by \(S\) with the discrete topology, and let \(G\) be its Pontryagin dual. Then \(G\) is compact and there exists a continuous homomorphism \(\varphi : \mathbb R \mapsto G\) with a dense image. Then \(\Gamma = \{\chi \circ \varphi : \chi \text{ is a character on } G\}\).\N\NProposition 1 shows that there exists a measure on \(G\) whose Fourier series equals the second equation above. Lemma 5 shows that if a Radon measure is translation bounded and its Fourier-Bohr transform \(\widehat \mu\) is supported on a discrete set then \(\mu\) is a Bohr almost periodic measure. We observe that this measure is the image of \(\mu \circ \varphi\) under the map induced by the canonical epimorphism of \(B(\mathbb R)\) onto \(G\). We now discuss Sections 3--8.\N\N\textbf{Traces of Radon Measures.} Assume \(\mathbb T^m\) is the \(m < \infty\) dimensional torus group and \(\varphi : \mathbb R \mapsto \mathbb T^m\) is a continuous homomorphism with a dense image. In Section 3 Meyer introduces the concept of a measure \(\sigma\) on\(\mathbb T^m\) being transverse to \(\varphi\) and shows that then there exists a unique Bohr almost periodic measure on \(\mathbb R\), which he calls the trace of \(\sigma\) and denotes by \(\sigma \circ \varphi\), for which \(\sigma\) is the measure on \(\mathbb T^m\) corresponding to \(\sigma \circ \varphi\) by Proposition 1.\N\NWe now generalize these concepts troduced by Meyer, based on Pontryagin duality, to show that they do not require that \(\Gamma\) be finitely generated so \(G\) need not be a torus group. Let \(H\) be a locally compact abelian group, let \(G\) be a compact group, and let \(\varphi : H \mapsto G\) be a continuous homomorphism with a dense image. Let \(dh\) and \(dg\) be Haar measures on \(H\) and \(G\) with \(\int_G dg = 1\). \(dh\) is uniquely determined up multiplicatio by a positve number. Let \(\mathcal M(H)\) and \(\mathcal M(G)\) be the Banach algebras of complex-valued Borel measures on \(H\) and \(G\) under convolution. These are the duals of \(C_b(H)\) and \(C(G)\). For a measure \(\mu\) and continuous functions we define \(<\mu,f>\) to ne the integral over \(H\) or \(G\) of \(f\) with respect to \(\mu\). \(\varphi\) induces a map \(\varphi^* : \mathcal M(H) \mapsto \mathcal M(G)\) defined by\N\[\N<\varphi^*(\mu),g> \, := \, <\mu \circ g \circ \varphi>, \ \ g \in C_b(G)\N\]\Nwhere \(g \circ \varphi \in C_b(H)\) is the composition of \(g\) and \(\varphi\). If \(f \in C_c(H)\) then \(fdh \in \mathcal M(H)\) hence \(\varphi^*(fdh) \in \mathcal M(G)\). Define \(\sigma \in \mathcal M(G)\) to be transverse to \(\varphi\) if for every \(f \in C_c(H)\), \(\sigma*\varphi^*(fdh)\) is absolutely continuous with respect to \(dg\) and its Radon-Nikodym derivative\N\[\N\widetilde f := \frac{d \sigma*\varphi^*(fdh)}{dg} \in C(G)\N\]\NThen the composition \(\widetilde f \circ \varphi\) is a Bohr almost periodic function on \(H\). The Banach-Alaoglu theorem implies that if \(f_j\) is an approximation of the identity, then \(\widetilde {f_j} \circ \varphi\) converges weakly to a Radon measure denote by \(\sigma \circ \varphi\) (not to be confused with a composition of fucntions. It is called the trace of \(\sigma\). It is easy to show that \(\widetilde f = f*(\sigma \circ \varphi)\). Furthermore \(\sigma\) is the measure on \(G\) in Proposition 1 nduced by the measure \(\sigma \circ \varphi\) on \(H\). This gives a bijection between transverse to \(\varphi\) measures on \(G\), and Bohr almost periodic measures on \(H\) whose spectrums are contained in \(\widehat \varphi(\widehat G) \subset \widehat H\). Here \(\, \widehat { }\, \) denotes the Pontryagin duality functor.\N\NIn Section 3 Meyer constructs concrete examples of measures on \(\mathbb T^m\) that are trasverse and not transverse to \(\varphi : \mathbb R \mapsto \mathbb T^n\). These examples give intuitive insight. Lemmas 6 and 7 and Theorem 2 describe geometric while Lemma 8 and Corollart 1 describe spectral properties of the correspondence described in the preceeding paragraph.\N\N\textbf{Sparse Crystalline Measures.} Parameterize \(\mathbb T\) by \(\mathbb R/\mathbb Z\). Let \(\phi : \mathbb T^{m-1} \mapsto \mathbb T\) be a \(C^\infty\) map where \(\phi\) is identified with a \(C^\infty\) function \(\phi : \mathbb R^{m-1} \mapsto \mathbb R\) satisfying \(\phi(x+k) = \phi(x) + q \cdot k\) for some \(q \in \mathbb Z^{m-1}\). Define \(U_\phi := \{(x,\phi(x)) : x \in \mathbb T^{m-1}\} \subset \mathbb T^m\) be its graph, and define \(\Phi : \mathbb T^{m-1} \mapsto \mathbb T^m\) by \(\Phi(x) := (x,\phi(x)\). Define \(\sigma_U \in \mathcal M(\mathbb T^m\) to be the image of Haar measure on \(\mathbb T^{m-1}\) under \(\Phi^*\). \(\sigma_U\) is an Ahern measure if \(\widehat {\sigma_U}(k) = 0\) for all \(k \in \mathbb Z^m\) unless all components of \(k\) are nonegative of all components are nonnegative. Assume that \(\alpha \in \mathbb R^m\) and define \(\varphi : \mathbb R \mapsto \mathbb T^m\) by \(\varphi(t) := t\alpha \mod \mathbb Z^m\). Clearly \(\varphi\) has a dense image iff \(\alpha\) has rationally independent components. Clearly \(\sigma)U\) is transverse to \(\varphi\) iff \(\alpha\) is transverse to \(U\) as defined in Definition 11. Theorem 3 shows that if \(\alpha\) is transverse to \(U\) and all componenst of \(\alpha\) are positive and \(\sigma_U\) is an Ahern measure then its trace \(\mu := \sigma_U \circ \varphi\) is a uniformly discrete Bohr almost periodic measure. Furthermore\N\[\N \mu = \sum_{k \in \mathbb Z^m} \widehat {\sigma_U}(k) \, \exp [2\pi i (k \cdot \alpha)t]\tag{2}\N\]\Nconvergence as a distribution so \(\mu\) is a crystalline measure since the support of \(\widehat \mu\) is contained in the set \(\{\alpha \cdot k\}\) where all components of \(k\) are nonnegative of all components of \(k\) are nonpositive and this set is discrete. A more detailed analysis shows that \(\mu\) is a Fourier quasicrystal.\N\N\textbf{Curved Model Sets.} The first sentence in Section 5 states ''The goal of this section is to bridge the gap between quaiscrystals, model sets, and crystalline measures.'' For \(\alpha\) irrational and \(s : \mathbb R \mapsto [0,1)\) the sawtooth function, the set \(\Lambda := \{k + s(k\alpha) : k \in \mathbb Z\}\) is a model set. The associated measure \(\sigma_\Lambda := \sum_{\lambda \in \Lambda} \delta_\lambda\) is Besicovitch but not Bohr almost periodic because \(s\) is not continuous. In contrast the set \(\{k + \sin(2\pi k\alpha) : k \in \mathbb Z\}\) is a curved model set. \textit{Y. Meyer} [``Global and local estimates on trigonometric sums'', Trans. Royal Norwegian Soc. Sci. Lett. 2, 1--25 (2018)] proved that \(\sigma_\Lambda\) is Bohr almost periodic and its Fourier transform is a dense Radon measure. He observed that if the \(\sin\) function is replaced by a suitable analytic function then \textit{P. Kurasov} and \textit{P. Sarnak} [J. Math. Phys. 61, No. 8, 083501, 13 p. (2020; Zbl 1459.05177)] proved that \(\sigma_\Lambda\) is a crystallline measure. General model sets have the form\N\[\N \Lambda := \varphi^{-1}(U)\tag{3}\N\]\Nwhere \(U\) is a flat subset of \(\mathbb T^m\) and \(\varphi : \mathbb R \mapsto \mathbb T^m\) is a continuous homomorphism with a dense image. Flat means that \(U = \Pi(K)\) where \(K\) is polytope concatined in an \((m-1)\)-dimensional affine subspace \(V \subset \mathbb R^m\), and \(\Pi : \mathbb R^m \mapsto \mathbb T^m\) is the canonical epimorphism, and \(U\) is transverse to \(\varphi\)and \(U\) has a boundary. Clearly \(U\) has no boundary iff \(U\) is a translate of an \((m-1)\)-dimensional torus sugbroup ot \(\mathbb T^m\) and then \(\Lambda = a\mathbb Z + b\) for some positive \(a\) and real \(b\) and \(\sigma_\Lambda\) is a trivial crystalline measure. Otherwise \(\mu\) is Besicivitch but nor Bohr almost periodic and \(\widehat \mu\) is not a Radon measure. If \(U\) is an \((m-1)\)-dimensional submanifold of \(\mathbb T^m\) transverse to \(\varphi\) then \(\Lambda\)is a curved model set. Then \(\sigma_\Lambda\) is Bohr almost periodic but usually not a crystalline measure. Theorem 5 shows that the trace of the measure \(\sigma_U\) is (2) is supported on \(\Lambda\) but is not equal to \(\sigma_\Lambda\).\N\N\textbf{Ahern Measure and Inner Functions} Meyer reviews standard facts about inner functions and Blasschke products. Theorem 4 shows that \(\sigma_U\) is an Ahern measure iff \(\exp [-2\pi i \phi]\) is an inner function is an inner function on \(\mathbb T^{m-1}\) when \(T\) is identified with the multiplicative group of complex numbers with modulus \(1\). Theorem 5 shows that if \(\exp [-2\pi i \phi]\) is an inner function and its graph \(U\) is transverse to \(\varphi\) and \(\sigma_U \in \mathcal M(\mathbb T^m)\) is the image of Haar measure under \(\Phi((x) := (x,\phi(x))\) then \(\sigma_U\) is transverse to \(\varphi\) and its trace \(\sigma_U \circ \varphi\) is a crystalline measure and\N\[\N \mu = \sum_{k \in \mathbb Z^m} \widehat {\sigma_U}(k) \, \exp [2\pi i (k \cdot \alpha)t]\tag{4}\N\]\N\N\textbf{The Set Constructed by Kurasov and Sarnak Is a Curved Model Set.} A trigonometric polynomial \(P\), e.g., a linear combination of exponential functions \(\exp(2\pi i \omega z)\) with real \(\omega\), is called real-rooted the set of its roots \(\Lambda \subset \mathbb R\) and then its divisor is the Radon measure\N\[\N\operatorname{div} P := \sum_{\lambda \in \Lambda} m(\lambda) \delta_\lambda\N\]\Nwhere \(m(\lambda)\) is the multiplicity of \(\lambda\). Section 7 shows that the crystalline measures construction by \textit{P. Kurasov} and \textit{P. Sarnak} [J. Math. Phys. 61, No. 8, 083501, 13 p. (2020; Zbl 1459.05177)] are more general than the construction using Ahern measures. Their construction has two parts. First, they prove that divisors of real-rooted trigonometric polynomials are crystalline measures and in fact are Fourier quasicrystals. They address the case where \(\Lambda\) is uniformly discrete and \(m(\lambda) = 1\), but their contour integration based proof extends to the general case. Secondly they construct a class of real-rooted trigonometric polynomials having the form\N\[\N P(t) = L(\exp(2\pi i \alpha_1 t),\dots,\exp(2\pi i \alpha_m t)\tag{5}\N\]\Nwhere \(L\) is a Lee-Yang polynomial and all \(\alpha_j > 0\). Such compositions are clearly real rooted trigonometric polynomials. Subsequent to the publication of this paper, \textit{L. Alon} et al. [J. Funct. Anal. 286, No. 2, Article ID 110226, 10 p. (2024; Zbl 1537.32004)] proved that every real-rooted trigonometric polynomial is described by (5). Alexander Olevsky and Alexander Ulanovsky [19] proved that all Fourier quasicrystals are divisors of real-rooted trigonometric polynomials. These two results show that the class of crystalline measures constructed by Kurasov and Sarnak are exactly the same as those constructed by Olevsky and Ulanovsky. This contradicts Meyer's following assertion (on page 403, lines 9--10): ``In [C. R., Math., Acad. Sci. Paris 358, No. 11--12, 1207--1211 (2020; Zbl 1456.42032)] \textit{A. Olevskii} and \textit{A. Ulanovskii} improved on the results of [\textit{Y. F. Meyer}, Bull. Hell. Math. Soc. 61, 11--20 (2017; Zbl 1425.42008)]. They elaborate a class of crystalline measures which is larger than the one which is constructed in [\textit{Y. F. Meyer}, Bull. Hell. Math. Soc. 61, 11--20 (2017; Zbl 1425.42008)].''\N\NLemma 9 uses a result of \textit{W. Rudin} and \textit{E. L. Stout} [J. Math. Mech. 14, 991--1005 (1965; Zbl 0147.11601)], which parameterizes all inner functions, to construct stable polynomials from inner functions. Meyer shows that one of the two examples of Fourier quasicrystals in [\textit{P. Kurasov} and \textit{P. Sarnak}, J. Math. Phys. 61, No. 8, 083501, 13 p. (2020; Zbl 1459.05177)] is constructed from an inner function while the other is not so their method is more general than the method based on Ahern measures. Theorems 6 and 7 describe constructions based on Ahern measures. The appendix gives a detailed computation of a trace.\N\NRemark. Positive crystalline measures on \(\mathbb R^n\) with uniformly discrete supports are Bohr almost periodic measures but the converse does not hold. \textit{W. M. Lawton} [J. Geom. Anal. 32, No. 2, Paper No. 60, 20 p. (2022; Zbl 1493.43003)] proves that all Bohr almost periodic measures on \(\mathbb R^n\) with uniformly discrete support and whose spectrums are contained in groups generated by \(m \geq 2\) elements are arise from curved model sets. This shows that that the concept of curved model sets has applications for more measure more general than crystalline measures on \(\mathbb R\).\N\N\textbf{Errata.}\N\begin{itemize}\N\item[1.] On pages 390 and 403 all references to [9] (written by Meyer) should be changed to reference [14] in this review (written by Olevskii and Ulanovskii)\N\item[2.] The contradiction on page 403 noted above should be observed.\N\end{itemize}\N\NFor the entire collection see [Zbl 1516.00008].
scientific article; zbMATH DE number 7914256