arXiv subjects
Publications and source records attributed to Emmanuel Cabanillas.
In this article, we propose a variant of the usual Ostrowski $α$-numeration (where $α$ is a real in [0, 1[) that codes integers (positive as well as negative) and reals of [0, 1[ (instead of [--$α$, 1--$α$[), so that for every integer n, n and {n$α$} have the same coding sequence. These coding sequences respect natural lexicographic orders and will be used to prove well known results on order properties of Kronecker sequences ({n$α$ -- $β$}) n .
In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like numeration of some rationals, simply related to entries a, b, d. So, we obtain quadratic complexity algorithms to compute these invariants. As a consequence, we will prove that, for these numerical semigroups, the type is always lower than the embedding dimension and deduce Wilf's property. We also consider the '' reverse problem'' : given J, a finite set of integers, we obtain an expression of all possible triplets (a, b, d), such that J is the set of minimal generators of (a,b) over d.