PROPORTIONALLY MODULAR DIOPHANTINE INEQUALITIES AND THEIR MULTIPLICITY

Abstract

Let I be an interval of positive rational numbers. Then the set S(I ) Æ T \N, where T is the submonoid of ¡ QÅ 0 ,Å ¢ generated by T , is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of aDiophantine inequality of the form ax mod b · cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S(I ) has a given multiplicity. We also characterize the numerical semigroups associated to these maximal intervals.