Let $A, B\subseteq \mathbb{ R}^2$ be finite, nonempty subsets, let $s\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines, $\ell_1,...,\ell_{ t},\ell'_1,...,\ell'_{ t}$, with $A\subseteq \bigcup_{ i=1}^{ t}\ell_i$ and $B\subseteq\bigcup_{ i=1}^{ t}\ell'_i$. Suppose $h_1(A,B)\geq s$. Then we show that: (a) if $||A|-|B||\leq s$ and $|A|+|B|\geq 4s^2-6s+3$, then $$|A+B|\geq (2-\frac 1 s)(|A|+|B|)-2s+1;$$ (b) if $|A|\geq |B|+s$ and $|B|\geq 2s^2-{ 7/2}s+{ 3/2}$, then $$|A+B|\geq |A|+(3-\frac 2 s)|B|-s;$$ (c) if $|A|\geq { 1/2}s(s-1)|B|+s$ and either $|A|> { 1/8}(2s-1)^2|B|-{ 1/4}(2s-1)+\frac{ (s-1)^2}{ 2(|B|-2)}$ or $|B|\geq \frac{ 2s+4}{ 3}$, then $$|A+B|\geq |A|+s(|B|-1).$$ 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.
Categories: math.CO math.NT