Title:  Circle Packings and a generalization of Steinhaus' theorem

Abstract: A classic theorem due to Steinhaus states that if A is a set of positive Lebesgue measure, then the difference set, A-A = {x-y: x,y in A}, has non-empty interior.  The result is proved, for instance, by the Lebesgue density theorem.
Later, Erdos and Oxtoby extend Steinhaus' result in the linear setting to show that if H is a sufficiently nice real-valued function on the plane, and if A and B are linear sets of positive Lebesgue measure, then H(A,B) = {(x,y) : x in A,   y in  B }  contains a non-empty open interval.
In a joint work with Karoly Simon, we extend Erdos and Oxtoby's result to families of functions and use this generalization to study a sum set problem.  Mainly, we show that if A is a sufficiently regular subset of the plane, then A+ S^1 has non-empty interior where S^1 denotes the circle in the plane.  Note that A+ S^1 is equal to union of all circles of radius one centered at points in A.  We discuss further directions and connections with the ultimate goal of understanding the interior of general sum sets A+ B where A and B can sets of fractal dimension.