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Atomic surfaces, tilings and coincidences II. Reducible case

Hiromi Ei, Shunji Ito, Hui Rao (2006)

Annales de l’institut Fourier

The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that...

Au bord de certains polyèdres hyperboliques

Marc Bourdon (1995)

Annales de l'institut Fourier

Le cadre de cet article est celui des groupes et des espaces hyperboliques de M.  Gromov. Il est motivé par la question suivante : comment différencier deux groupes hyperboliques à quasi-isométrie près ? On illustre ce problème en détaillant un exemple de M. Gromov issu de Asymptotic invariants for infinite groups. On décrit une famille infinie de groupes hyperboliques, deux à deux non quasi-isométriques, de bord la courbe de Menger. La méthode consiste à étudier leur structure quasi-conforme au...

Ausbohrung von Rhomben

B. WEISSBACH (1977)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

Average decay of Fourier transforms and geometry of convex sets.

Luca Brandolini, Marco Rigoli, Giancarlo Travaglini (1998)

Revista Matemática Iberoamericana

Let B be a convex body in R2, with piecewise smooth boundary and let ^χB denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical Lp averages of ^χB and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.

Balancing vectors and convex bodies

Wojciech Banaszczyk (1993)

Studia Mathematica

Let U, V be two symmetric convex bodies in ℝ n and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors u 1 , . . . , u n ∈ U such that, for each choice of signs ε 1 , . . . , ε n = ± 1 , one has ε 1 u 1 + . . . + ε n u n ∉ r V where r = ( 2 π e 2 ) - 1 / 2 n 1 / 2 ( | U | / | V | ) 1 / n . Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence ( u n ) such that the series ∑ n = 1 ∞ ε n u π ( n ) is divergent for any choice of signs ε n = ± 1 and any permutation π of indices.

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