Symmetric Modular Diophantine Inequalities

In this paper we study and characterize those Diophantine inequalities axmod b ≤ x whose set of solutions is a symmetric numerical semigroup. Given two integers a and b with b = 0 we write a mod b to denote the remainder of the division of a by b. Following the notation used in [8], a modular Diophantine inequality is an expression of the form ax mod b ≤ x. The set S(a, b) of integer solutions of this inequality is a numerical semigroup, that is, it is a subset of N (the set of nonnegative integers) closed under addition, 0 ∈ S(a, b) and such that N\S(a, b) has finitely many elements. We say that a numerical semigroup ismodular if it is the set of solutions to a modular Diophantine inequality. As shown in [8], not every numerical semigroup is of this form. If S is a numerical semigroup, then the greatest integer not in S is the Frobenius number of S, denoted by g(S). The numerical semigroup S is symmetric (see [1]) if x ∈ Z\S implies g(S)−x ∈ S (Z is the set of integers). This kind of semigroup has been widely studied and characterized in the literature (see, for instance, [2, 4, 6]). We will say that the inequality ax mod b ≤ x is symmetric if S(a, b) is a symmetric numerical semigroup. It is well known (see, for instance, [7]) that every numerical semigroup S is finitely generated and thus there exist positive integers n1, . . . , np such that S = 〈n1, . . . , np〉 = {a1n1 + · · ·+ apnp | a1, . . . , ap ∈ N}. If no proper subset of {n1, . . . , np} generates S, then we say that this set is a minimal system of generators of S. Minimal systems of generators always exist and are unique (see [7]); the cardinality of a minimal system of generators of S is known as the embedding dimension of S, denoted here by e(S). Clearly, the inequality ax mod b ≤ x has the same integer solutions as the inequality (a mod b)x mod b ≤ x. Thus we may assume (and in fact we will) that a, b ∈ N and a < b. Note that S(0, b) = N is trivially symmetric. Throughout this paper (and unless otherwise stated) we will assume that a and b are positive integers, and that d = gcd{a, b} and d′ = gcd{a−1, b} (gcd stands for greatest common divisor). In Proposition 4, we will show that S(a, b) is symmetric if and only if S(a, b) = 〈b/d, b/d′, d+ d′〉. Theorem 5 characterizes those pairs (a, b) Received by the editors April 8, 2004 and, in revised form, June 21, 2005. 2000 Mathematics Subject Classification. Primary 20M14. The author was supported by the project BFM2000-1469 and thanks P. A. Garćıa-Sánchez for his comments and suggestions. c ©2006 American Mathematical Society