Let \ (A Zᵈ\) be a finite set. It is known that the sumset \ (NA\) has predictable size (\ (NA = PA (N) \) for some \ (PA (X) QX\) ) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once \ (N\) is larger than some threshold. In previous work, the first effective bounds for both of these thresholds were established, for an arbitrary set \ (A\). In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known. Mathematics Subject Classifications: 11P21, 05B10, 11B13, 11P70, 05A16Keywords: Sumsets, Set addition, Khovanskii polynomial, Structure Theorem, Explicit Bounds
Granville et al. (Fri,) studied this question.