Let \(A \subset {\mathbb {Z}}^d\) be a finite set. It is known that NA has a particular size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in {\mathbb {Q}}[X]\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results were only previously known in the special cases when \(d=1\), when the convex hull of A is a simplex or when \(\vert A\vert = d+2\) Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.

设\(A \subset {\mathbb {Z}}^d\)为有限集。已知NA具有特定的大小（对于 {\mathbb {Q}}[X]\) 中的某些\(P_A(X) \) ， NA 具有特定的大小（ \(\vert NA\vert = P_A(N) \)）和结构（一旦N大于某个阈值，锥体中的所有格点（某些特殊集合除外） 。在本文中，我们给出了任意A的该阈值的第一个有效上限。这种显式结果以前仅在以下特殊情况下已知：\(d=1\) 、 A的凸包是单纯形或\(\vert A\vert = d+2\) Curran 和 Goldmakher（离散肛门） . 论文第 27 号，2021），我们改进的结果。