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A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated by $A$ and if $n \ge R$ then $n$ is in the semigroup generated by $A$.

Are there any references to a higher dimensional version of this result?

The version I want goes like this.

  • Take a finite subset $U$ of $\mathbb{N}^n$. Let $C_U$ be the smallest closed cone in $\mathbb{R}^n$ containing $U$, i.e. all non-negative real linear combinations of $U$. Let $G_U$ be the subgroup of $\mathbb{N}^n$ generated by $U$, i.e. all integer linear combinations. Let $S_U$ be the subsemigroup generated by $U$, i.e. all non-negative integer linear combinations. Then there exists $R>0$ such that for every $v \in G_U$, if the ball around $v$ of radius $R$ is contained in $C_U$ then $v \in S_U$.
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    $\begingroup$ I don't have any literature in front of me. The right buzzword is affine semigroups because that is what people who study them call them. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 2, 2012 at 19:58
  • $\begingroup$ emis.de/journals/SC/2002/6/pdf/smf_sem-cong_6_43-127.pdf seems to have some info on affine semigroups, the group they span and the polyhedral cone they span. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 2, 2012 at 20:10
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    $\begingroup$ A higher dimensional analogue of the result you mention in dimension 1 can be found as Exercise 7.15 of Miller and Sturmfels combinatorial commutative algebra book but it is not quite what you want, I think. Let $N=C_U\cap G_U$. Then $N$ is a finitely generated semigroup and there exists according to exercise 7.15 an element $a$ of $S_U$ with $a+N\subseteq S_U$. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 2, 2012 at 20:42
  • $\begingroup$ That would do it. Take $R > |a|$. If $B_R(v) \subset C_U$ then $v-a \in C_U \cap G_U = N$ so $v \in a+N \subset S_U$. Great! $\endgroup$
    – Lee Mosher
    Commented Aug 2, 2012 at 21:07
  • $\begingroup$ Glad to be of help. Perhaps we should copy this into the answer box so that the software knows it is answered? $\endgroup$
    – Benjamin Steinberg
    Commented Aug 3, 2012 at 0:38

1 Answer 1

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This combines comments of myself and Lee Mosher.

Exercise 7.15 of Miller and Sturmfels combinatorial commutative algebra book proves the following. Let $N=C_U\cap G_U$. Then $N$ is a finitely generated semigroup and there exists according to exercise 7.15 an element $a$ of $S_U$ with $a+N\subseteq S_U$. Now take $R>|a|$. If $B_R(v)\subseteq C_U$ then $v−a\in C_U\cap G_U=N$ so $v\in a+N\subseteq S_U$.

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  • $\begingroup$ By the way, my original purpose in asking this question was to provide a reference for Thurston's paper "Entropy in Dimension 1"; see Lemma 11.6 of that paper. The paper has now been published in "Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday". $\endgroup$
    – Lee Mosher
    Commented Jul 17, 2014 at 16:49

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