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Fourier analysis, linear programming, and densities of distance avoiding sets in ℝ n

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

We derive new upper bounds for the densities of measurable sets in ℝ n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2 , ⋯ , 24 . This gives new lower bounds for the measurable chromatic number in dimensions 3 , ⋯ , 24 . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...

Generalized Whitney partitions

Michał Rams (2000)

Fundamenta Mathematicae

We prove that the upper Minkowski dimension of a compact set Λ is equal to the convergence exponent of any packing of the complement of Λ with polyhedra of size not smaller than a constant multiple of their distance from Λ.

Inhomogeneous extreme forms

Mathieu Dutour Sikirić, Achill Schürmann, Frank Vallentin (2012)

Annales de l’institut Fourier

G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension  6 and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.

John Horton Conway (1937–2020)

Petr Stehlík, Václav Vopravil (2020)

Pokroky matematiky, fyziky a astronomie

Angloamerický matematik John Horton Conway byl všestrannou a charismatickou postavou, která významně ovlivnila teorie čísel, grup, her, uzlů, dynamických systémů i rekreační matematiku. Proslul svéráznou povahou i nekonvenčním přístupem k řešení problémů. Tento článek shrnuje stručně jeho neobvyklou životní cestu a představuje čtyři vybrané oblasti z jeho bohaté tvorby: nadreálná čísla, teorii kombinatorických her, hru života a klasifikaci sporadických grup.

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