Take the 2-minute tour ×  Is there a characterization of measure
zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$
contains interior? Here $A+A$ is the set of points $\{ x+y \mid
x, y\in A \}$.  Is it true that if the convex hull of the
connected component of $A$ contains interior then so does $A+A$?
share|improve this question add comment  2 Answers  up vote 0
down vote accepted  Regarding Question #1.  There's one obvious
dimensional obstruction (for example consider the Hausdorff
dimension of A).  There is some research about it in the
1-dimensional case, such as the well-known theorem that K+K
contains an interval, and related conjectures and works by
Pallis, Furstenberg and Yoccoz. Even this one-dimensional theory
is not complete as far as I know, so characterizing such a
statement in larger dimensions would seem improbable now.  If
say your set A is the product of two sets $A=A_{x} \times
A_{y}$, such that $A_{x}+A_{x}$ contains an interval, and
$A_{y}+A_{y}$ contains an interval, then o