Properties of two dimensional sets with small sumset

Let $A,B\subseteq {\mathbb{R}}^2$ be finite, nonempty subsets, let $s\ge 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines, ${\mathrm{\ell }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\ell }}_t,{\mathrm{\ell }}_1^{\mathrm{\prime }},\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\ell }}_t^{\mathrm{\prime }}$, with $A\subseteq \bigcup_{i=1}^{t}\ell_i$and$B\subseteq\bigcup_{i=1}^{t}\ell’_i$\mathrm{.}Suppose$h_1(A,B)\geq s$. Then we show that: (a) if $\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }-\mathrm{\mid }B\mathrm{\mid }\mathrm{\mid }\le s$ and $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }\ge 4s^2-6s+3$, then $$\mathrm{\mid }A+B\mathrm{\mid }\ge (2-\frac{1}{s})(\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid })-2s+1;$$ (b) if $\mathrm{\mid }A\mathrm{\mid }\ge \mathrm{\mid }B\mathrm{\mid }+s$ and $\mathrm{\mid }B\mathrm{\mid }\ge 2s^2-7\mathrm{/}2s+3\mathrm{/}2$, then $$\mathrm{\mid }A+B\mathrm{\mid }\ge \mathrm{\mid }A\mathrm{\mid }+(3-\frac{2}{s})\mathrm{\mid }B\mathrm{\mid }-s;$$ © if $\mathrm{\mid }A\mathrm{\mid }\ge 1\mathrm{/}2s(s-1)\mathrm{\mid }B\mathrm{\mid }+s$ and either $|A|> {1/8}(2s-1)^2|B|-{1/4}(2s-1)+\frac{(s-1)^2}{2(|B|-2)}$or$|B|\geq \frac{2s+4}{3}$,then$$\mathrm{\mid }A+B\mathrm{\mid }\ge \mathrm{\mid }A\mathrm{\mid }+s(\mathrm{\mid }B\mathrm{\mid }-1)\mathrm{.}$$Thisextendsthe2-dimensionalcaseoftheFreiman$2^d$–Theorem to distinct sets $A$ and $B$, and, in the symmetric case $A=B$, improves the best prior known bound for $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }$ (due to Stanchescu, and which was cubic in $s$) to an exact value. As part of the proof, we give general lower bounds for two dimensional subsets that improve the 2-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, and that generalize the 2-dimensional case of the Brunn-Minkowski Theorem.