Geometric study of the set � β of beta - integers with β a Perron number , a β - number and

We investigate in a geometrical way the sieving process of 021 β 3 for obtaining the Delone set 0 β 4 051 β 3 of β integers where β is a Perron number in the context of linear asymptotic invariants associated with a canonical inductive system constructed from β . When β is a Pisot number, we exhibit a canonical cut-and-project scheme, a model set associated with 0 β and so prove that it is a Meyer set. We show how to lift up the elements of 0 β to a subset 687:9 of the lattice 0 m ; m < degree β = , lying about the dominant eigenspace of the companion matrix of β . We deduce from this linearized version of 0 β (i) the existence of a finite number of elements g1, g2, >?>@> , gη A 0 B β of small norm such that the semi-group C 1 g1, g2, >@>?> , gη 3 contains 0 B β except possibly a finite number of elements close to the origin, (ii) an upper bound for the integer 7 taking place in the relation