Improved stability for the size and structure of iterated sumsets in \(\mathbb{Z}^d\)

Let \(A \subset \mathbb{Z}^d\) be a finite set. It is known that the sumset \(NA\) has predictable size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in \mathbb{Q}[X]\)) 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