Shifts of generators and delta sets of numerical monoids. (Q2923340)

Let \(a_1,\ldots,a_t\) be a set of \(t\) positive integers. A factorization of \(s\) in terms of \(a_1,\ldots,a_t\) is an expression of the form \(s=z_1a_1+\cdots+z_ta_t\), with \(z_1,\ldots,z_t\) nonnegative integers. The length of this factorization is \(\sum_{i=1}^tz_i\). The set of all lengths of factorizations of \(s\) with respect to \(a_1,\ldots,a_t\) is finite, and so we can write it as \(\{l_1<\cdots<l_k\}\). The Delta set of factorizations of \(s\) is defined as \(\Delta(s)=\{l_2-l_1,l_3-l_2,\ldots,l_k-l_{k-1}\}\).NEWLINENEWLINE For \(S\) the monoid generated by \(\{a_1,\ldots,a_t\}\) (that is, the set of integers \(s\) having at least one factorization in terms of \(a_1,\ldots,a_t\)), we define \(\Delta(S)=\bigcup_{s\in S}\Delta(s)\). For \(t>1\), \(\Delta(S)\) is not empty, and so the simplest Delta set we can obtain is a singleton.NEWLINENEWLINE Write the set \(\{a_1,\ldots,a_t\}\) as \(\{n,n+r_1,\ldots,n+r_{t-1}\}\), with \(r_1<\cdots<r_{t-1}\). The authors prove that fixed \(r_1,\ldots,r_{t-1}\), for \(n\) larger than a constant \(N\) (depending on \(r_{t-1}\) and \(t\)), the set \(\Delta(S)\) is a singleton. This bound is improved for embedding dimension three, that is \(t=2\) and no \(a_i\) has a factorization in terms of the rest. These results have another reading. For \(a_1<\cdots<a_t\), if \(a_t\) is close enough to \(a_1\), then we obtain a monoid with the simplest possible Delta set.