The sum-product problem for small sets

For $A\subseteq \mathbb{R}$, let $A+A=\{a+b:a,b\in A\}$ and $AA=\{ab:a,b\in A\}$. For $k\in \mathbb{N}$, let $SP(k)$ denote the minimum value of $\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}$ over all $A\subseteq \mathbb{N}$ with $\mathrm{\mid }A\mathrm{\mid }=k$. Here we establish $SP(k)=3k-3$ for $2\le k\le 7$, the $k=7$ case achieved for example by $\{1,2,3,4,6,8,12\}$, while $SP(k)=3k-2$ for $k=8,9$, the $k=9$ case achieved for example by $\{1,2,3,4,6,8,9,12,16\}$. For $4\le k\le 7$, we provide two proofs using different applications of Freiman’s $3k-4$ theorem; one of the proofs includes extensive case analysis on the product sets of $k$-element subsets of $(2k-3)$-term arithmetic progressions. For $k=8,9$, we apply Freiman’s $3k-3$ theorem for product sets, and investigate the sumset of the union of two geometric progressions with the same common ratio $r>1$, with separate treatments of the overlapping cases $r\mathrm{\ne }2$ and $r\ge 2$.