Proportionally modular diophantine inequalities and their multiplicity

Let I be an interval of positive rational numbers. Then the set S (I) = T ∩ ℕ, where T is the submonoid of ℚ₀⁺, +) 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 a Diophantine 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 with these maximal intervals.