Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in ⁿ which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on ^s and ⁿ , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map and a constant C such that ||ϕ (x) - (x)|| for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | η; and if X is a Meyer set, then all algebraic conjugates | η ' | 1. Received May 9, 1997, and in revised form March 5, 1998.     

Substitution Delone Sets

   Abstract. Substitution Delone set families are families of Delone sets X =(X ₁ , . . ., X _n ) which satisfy the inflation functional equation

Multiregular Point Systems

This paper gives several conditions in geometric crystallography which force a structure X in R ⁿ to be an ideal crystal. An ideal crystal in R ⁿ is a finite union of translates of a full-dimensional lattice. An (r,R) -set is a discrete set X in R ⁿ such that each open ball of radius r contains at most one point of X and each closed ball of radius R contains at least one point of X . A multiregular point system X is an (r,R) -set whose points are partitioned into finitely many orbits under the symmetry group Sym(X) of isometries of R ⁿ that leave X invariant. Every multiregular point system is an ideal crystal and vice versa. We present two different types of geometric conditions on a set X that imply that it is a multiregular point system. The first is that if X ``looks the same' when viewed from n+2 points { y <sub>i</sub> : 1 i n + 2 } , such that one of these points is in the interior of the convex hull of all the others, then X is a multiregular point system. The second is a ``local rules' condition, which asserts that if X is an (r,R) -set and all neighborhoods of X within distance ρ of each x∈X are isometric to one of k given point configurations, and ρ exceeds CRk for C = 2(n ² +1) log ₂ (2R/r+2) , then X is a multiregular point system that has at most k orbits under the action of Sym(X) on R ⁿ . In particular, ideal crystals have perfect local rules under isometries. Received September 13, 1996, and in revised form September 27, 1996, February 6, 1997, and May 7, 1997.     

Asymptotic Geometry and Growth of Conjugacy Classes of Nonpositively Curved Manifolds

Let X be a Hadamard manifold and Γ⊂Isom(X) a discrete group of isometries which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the limit set of Γ in order to derive a new asymptotic estimate for the growth rate of closed geodesics in not necessarily compact or finite volume manifolds. Mathematics Subject Classifications (2000): 20E45, 53C22, 37F35     

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ ^d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ² and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

A new characterization of group action-based perfect nonlinearity

The left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X.     

Delone Sets in ℝ<Superscript>3</Superscript> with 2<Emphasis Type="Italic">R</Emphasis>-Regularity Conditions

A regular system is the orbit of a point with respect to a crystallographic group. The central problem of the local theory of regular systems is to determine the value of the regularity radius, which is the least number such that every Delone set of type (r,R) with identical neighborhoods/clusters of this radius is regular. In this paper, conditions are described under which the regularity of a Delone set in three-dimensional Euclidean space follows from the pairwise congruence of small clusters of radius 2R. Combined with the analysis of one particular case, this result also implies the proof of the “10R-theorem,” which states that if the clusters of radius 10R in a Delone set are congruent, then this set is regular.     

Reconstructing Finite Sets of Points in Rup to Groups of Isometries

We prove reconstruction results for finite sets of points in the Euclidean spaceRⁿthat are given up to the action of groups of isometries that contain all translations and for which the origin has a finite stabilizer.     

On Subsets with Small Product in Torsion-Free Groups

G be a nonabelian torsion-free group. Let C be a finite generating subset of G such that . We prove that, for all subsets B of G with , we have . In particular, a finite subset X with cardinality satisfies the inequality if and only if there are elements , such that the following two conditions hold: (i) . (ii) where . Received: October 13, 1997/Revised: Revised August 18, 1998     

Mappings preserving two hyperbolic distances

Suppose that X is the set of points of a hyperbolic geometry of finite or infinite dimension , and that is a fixed real number and N>1 a fixed integer. Let be a mapping such that for every if h(x, y)=, then h(f,(x),f,(y)) , and if h(x,y) = N, then h(f,(x),f,(y)) , where h,(p,q) designates the hyperbolic distance of p,q . Then f is an isometry of X. Note that there is no regularity assumption on f, like continuity or even differentiability. Moreover, we present an example showing that the assumption that one fixed distance > 0 is preserved does not characterize hyperbolic isometries. Received 17 February 2000.     

ACCP Rises to the Polynomial Ring if the Ring has Only Finitely Many Associated Primes

Abstract

We show for a commutative ring R with unity: If R satisfies the ascending chain condition on principal ideals (accp) and has only finitely many associated primes, then for any set of indeterminates X the polynomial ring R[X] also satisfies accp. Further we show that accp rises to the power series ring R[[X]] if R satisfies accp and the ascending chain condition on annihilators.     

Any Banach space has an equivalent norm with trivial isometries

For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ _G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ _G ) is isomorphic toG × {−1, + 1}.     

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R _Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ˉρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E ₀ of , then defines a smooth l-adic analytic variety, near the trivial lift ρ₀ of ˉρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R _Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ˉρ odd, using ideas of Coleman, Gouvêa and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ₀, provided this lift satisfies the Artin conjecture and E ₀ satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R _Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case. Received: 12 December 1997 / Revised version: 5 February 1998     

Size direction games over the real line. III

We prove, using the continuum hypothesis, thatD (the direction player) has a winning strategy in {ie442-1} for some uncountableX, and that there is an uncountableX which intersects each perfect nowhere-dense set of reals in a countable set such thatD does not win in {ie442-2} for everya. We also give another proof to the fact that Γ^S (X) is a win forD is countable.     

Séries de poincaré des groupes géométriquement finis

In this paper, we study the behaviour of the Poincaré series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

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Durant la rédaction de cet article, M. Peigné a bénéficié d'un détachement au Centre National de la Recherche Scientifique, URA 305.     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.