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On the closedness of approximation spectra

Jouni Parkkonen, Frédéric Paulin (2009)

Journal de Théorie des Nombres de Bordeaux

Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

For a real number $q > 1$ and a positive integer $m$ , let $Y_{m} (q) : = \{\sum_{i = 0}^{n} ϵ_{i} q^{i} : ϵ_{i} \in \{0, \pm 1, ..., \pm m\}, n = 0, 1, ...\}$ . In this paper, we show that $Y_{m} (q)$ is dense in $ℝ$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

On the topology of sums in powers of an algebraic number

Nikita Sidorov, Boris Solomyak (2011)

Acta Arithmetica

Simultaneous Approximation to Algebraic Numbers by Elements of a Number Field.

Wolfgang M. Schmidt (1975)

Monatshefte für Mathematik

Some remarks on the Approximation of Complex Numbers by Algebraic Numbers

Michael Laurent (1999)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Sur des hauteurs alternatives. II

Patrice Philippon (1994)

Annales de l'institut Fourier

Nous complétons l’interprétation géométrique de [P2]1991:726;1401792m:11061 pour les hauteurs locales archimédiennes et les distances projectives de [P1]88h11048. On montre comment ceci conduit à une taille (telle que définie dans [P3]) sur les anneaux de coordonnées de variétés projectives. On définit aussi des notions de maison et taille pour les extensions de type fini de $Q$ .