# Khovanskii's Theorem

September 2, 2022

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### Abstract

We formalise the proof of an important theorem in additive combinatorics due to Khovanskii, attesting that the cardinality of the set of all sums of $n$ many elements of $A$, where $A$ is a finite subset of an abelian group, is a polynomial in $n$ for all sufficiently large $n$. We follow a proof due to Nathanson and Ruzsa as presented in the notes “Introduction to Additive Combinatorics” by Timothy Gowers for the University of Cambridge.