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Ultrafilter extensions of asymptotic density

Jan Grebík (2019)

Commentationes Mathematicae Universitatis Carolinae

We characterize for which ultrafilters on $ω$ is the ultrafilter extension of the asymptotic density on natural numbers $σ$ -additive on the quotient boolean algebra $𝒫 (ω) / d_{𝒰}$ or satisfies similar additive condition on $𝒫 (ω) / fin$ . These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name $A P$ (null) and $A P$ (*). We also present a characterization of a $P$ - and semiselective ultrafilters...

Un théorème de finitude

Yvette Amice (1964)

Annales de l'institut Fourier

Démonstration élémentaire de la finitude de l’ensemble de type $H$ associé à une suite de densité uniforme extérieure non nulle.

Unique representation bases for the integers

Melvyn B. Nathanson (2003)

Acta Arithmetica

Untersuchungen über den Summenbegriff in der additiven Zahlentheorie

Hans-Heinrich Ostmann (1947/1949)

Mathematische Annalen

Using Bonse's inequality to find upper bounds on prime gaps.

Betts, Robert J. (2007)

Journal of Integer Sequences [electronic only]

Van der Corput sets in $ℤ^{d}$

Vitaly Bergelson, Emmanuel Lesigne (2008)

Colloquium Mathematicae

In this partly expository paper we study van der Corput sets in $ℤ^{d}$ , with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for d = 1 by Kamae and M. Mendès France and by Ruzsa, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some...

Verfeinerte Lösung der asymptotischen Dichtenaufgabe.

Hans-Heinrich Ostmann (1950)

Journal für die reine und angewandte Mathematik

Weak forms of Mann's density theorem extended to sets of lattice points

Betty Kvarda Garrison (1971)

Acta Arithmetica

Weighted uniform densities

Rita Giuliano Antonini, Georges Grekos (2007)

Journal de Théorie des Nombres de Bordeaux

We introduce the concept of uniform weighted density (upper and lower) of a subset $A$ of $ℕ^{*}$ , with respect to a given sequence of weights $(a_{n})$ . This concept generalizes the classical notion of uniform density (for which the weights are all equal to 1). We also prove a theorem of comparison between two weighted densities (having different sequences of weights) and a theorem of comparison between a weighted uniform density and a weighted density in the classical sense. As a consequence, new bounds for the...