Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 149 / 3

Composition and inversion in the Affine group of the complex line areThe mapis therefore a surjective group homomorphism with kernelIts target is Abelian, so $[G,G]\subseteq N$. On the other hand, the stated computation givesFixing any $a\ne 1$ and varying $b$ produces every translation. Thus $N\subseteq [G,G]$, and hence

The Solymosi sum-product theorem over the complex numbers states that if finite $U,V,W\subseteq \mathbb{C}$ satisfy $U\ne \{0\}$ and $W\ne \{0\}$, then

Write $N=[G,G]$ and identify it with $(\mathbb{C},+)$ as in part (a). If every two members of $A$ commuted, then $⟨A⟩$ would be Abelian, contrary to hypothesis. Thus some commutator of two members of $A$ is a nonidentity translation in $A^4\cap N$. Consequently the translation-coordinate setcontains both zero and a nonzero element.

The identityshows that $T\pi (A)$ is contained in the translation coordinates of $A^6\cap N$, while $T+T$ is contained in those of $A^8\cap N$. The intersection of an approximate group power with a subgroup therefore givesApply the Solymosi sum-product theorem over the complex numbers with $U=V=T$ and $W=\pi (A)$. Since $1\in \pi (A)$, its hypotheses hold, andCancelling $|T{|}^2$ and absorbing the absolute constant proves

Choose one representative from $A$ above each point of $\pi (A)$ and collect them in $X$. Part (c) gives $|X|\le K^{O(1)}$. If $a\in A$ has the same image as $x\in X$, then $x^{-1}a\in A^2\cap N$, soThis is the required covering of $A$ by at most $K^{O(1)}$ left cosets of the abelian translation subgroup $N$.

The set $B=A^2\cap N$ is a $K^{O(1)}$-approximate group by the intersection of an approximate group power with a subgroup. Apply the Freiman-Green-Ruzsa theorem inside $N\cong (\mathbb{C},+)$. Because the additive group of the complex numbers is a torsion-free group, the finite subgroup part is trivial, so there is an abelian progression $P$ withSince $|B|\le |A^2|\le K|A|$, enlarging the implicit constant gives