Properties of two dimensional sets with small sumset

Abstract

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, ${\ell }_1,...,{\ell }_t,{\ell }_1^{\prime },...,{\ell }_t^{\prime }$, with $A\subseteq {\bigcup }_{i=1}^t{\ell }_i$ and $B\subseteq {\bigcup }_{i=1}^t{\ell }_i^{\prime }$. Suppose $h_1(A,B)\ge s$. Then we show that: (a) if $||A|-|B||\le s$ and $|A|+|B|\ge 4s^2-6s+3$, then $$|A+B|\ge (2-\frac{1}{s})(|A|+|B|)-2s+1;$$ (b) if $|A|\ge |B|+s$ and $|B|\ge 2s^2-7/2s+3/2$, then $$|A+B|\ge |A|+(3-\frac{2}{s})|B|-s;$$ (c) if $|A|\ge 1/2s(s-1)|B|+s$ and either $|A|>1/8(2s-1{)}^2|B|-1/4(2s-1)+\frac{(s-1{)}^2}{2(|B|-2)}$ or $|B|\ge \frac{2s+4}{3}$, then $$|A+B|\ge |A|+s(|B|-1).$$ This extends the 2-dimensional case of the Freiman $2^d$--Theorem to distinct sets $A$ and $B$, and, in the symmetric case $A=B$, improves the best prior known bound for $|A|+|B|$ (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.