Ehrhart theory and an explicit version of Khovanskii's theorem

Abstract: A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. However, neither the polynomial nor what sufficiently large means are understood in general. We use Ehrhart theory to give a new proof of Khovanskii's theorem when $A \subset \mathbb{Z}^d$ that gives new insights into the growth of the cardinality of sumsets. Our approach allows us to obtain explicit formulae for $|hA|$ whenever $A \subset \mathbb{Z}^d$ contains $d + 2$ points that are valid for \emph{all} $h$. In the case that the convex hull $\Delta_A$ of $A$ is a $d$-dimensional simplex, we can also show that $|hA|$ grows polynomially whenever $h \geq \text{vol}(\Delta_A) \cdot d! - |A| + 2$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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