Monotone sets and growth in the Hypercube

Is the known definition for a monotone set in the hyperplane is : a set $A \in ${$0,1$}$^n$ such that for every $a \in A$ if $b < a$ so $b \in A$?
Given all hamming balls and all boxes (vector subspaces) in {$0,1$}$^n$. Can I construct a genereal set $A$ in the hypercube, as a minkowski sum of balls and boxes, that is defined uniquely?
What is known about the sequence $|A|,|A+B|,|A+2B|,\ldots$ for A a general set in hypercube, and B is the hamming ball of radius 1?

flag	asked Dec 14 at 13:01 ak47 29●3

	1) Yes; 2) No - consider the set {(1,1,1,1,1)} it's not the Minkowski sum of anything; 3) If $A$ is a singleton, the cardinalities are known. Similarly if $A$ is $2k$-discrete, then the $\|A+iB\|$ are known for $i\le k$. – Anthony Quas Dec 14 at 17:09
	Can you please write the definitions of $2k$-discrete sets and singleton sets (or reference me to these definitions)? thanks – ak47 Dec 15 at 15:04
	ok, got you. it's quite trivial :) – ak47 Dec 15 at 17:02