Matemática e Estatística | Artigos em revistas nacionais / Papers in national journals
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- Frisos imperfeitos de números inteirosPublication . Bessa, Mário; Carvalho, MariaA positive density ensures not only that the set is infinite but also that the arrangement of its elements is such that it is possible to find inside it highly symmetric and arbitrarily long blocks of equidistant points. For example, although this is not a necessary condition [1, 4], a set N with positive density must include arbitrarily long arithmetic progressions [3, 2]. That is, given a positive integer k, there are positive integers a,b such that a+jb ∈ N , for all j ∈ {0,...,k}. However, no information is given about the ratio b in the arithmetical progression, and surely not all choices of b are suitable for a fixed set. Nevertheless, a positive density set has to contain, up to an arbitrarily small error, a scaled copy of any finite block of rational numbers of [0, 1], for all scales not bigger than a specified bound.
- A note on reversibility and Pell equationsPublication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.This note concerns hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. Due to the low dimension, we will be able to associate the reversibility with a generalized Pell equation from whose set of solutions we will infer further information. Additionally, we will show that reversibility is a rare feature and will characterize the generic setting.