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On a generalization of the Selection Theorem of Mahler

Gilbert Muraz, Jean-Louis Verger-Gaugry (2005)

Journal de Théorie des Nombres de Bordeaux

The set 𝒰 𝒟 r of point sets of ℝ n , n ≥ 1 , having the property that their minimal interpoint distance is greater than a given strictly positive constant r > 0 is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset 𝒰 𝒟 r , f ⊂ 𝒰 𝒟 r of the finite point sets is compatible with the restriction of this topology to 𝒰 𝒟 r , f . We show that its subsets of Delone sets of given constants in ℝ n , n ≥ 1 , are compact. Three (classes of) metrics, whose one of crystallographic nature,...

On a question of Schmidt and Summerer concerning 3 -systems

Johannes Schleischitz (2020)

Communications in Mathematics

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper 3 -system ( P 1 , P 2 , P 3 ) with the property ϕ ¯ 3 = 1 . In fact, our method generalizes to provide n -systems with ϕ ¯ n = 1 , for arbitrary n ≥ 3 . We visualize our constructions with graphics. We further present explicit examples of numbers ξ 1 , ... , ξ n - 1 that induce the n -systems in question.

On integer points in polygons

Maxim Skriganov (1993)

Annales de l'institut Fourier

The phenomenon of anomaly small error terms in the lattice point problem is considered in detail in two dimensions. For irrational polygons the errors are expressed in terms of diophantine properties of the side slopes. As a result, for the t -dilatation, t → ∞ , of certain classes of irrational polygons the error terms are bounded as ≪ ℓ n q t with some q > 0 , or as ≪ t ϵ with arbitrarily small ϵ > 0 .

On systems of linear inequalities

Masami Fujimori (2003)

Bulletin de la Société Mathématique de France

We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.

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