Fractals and the monadic second order theory of one successor

We show that if X is virtually any classical fractal subset of R-n, then (R,<,+,X) interprets the monadic second-order theory of (N,+1). This result is sharp in the sense that the standard model of the monadic second-order theory of (N,+1) is known to interpret (R,<,+,X) for various classical fractals X including the middle-thirds Cantor set and the Sierpinski carpet. Let X subset of R-n be closed and nonempty. We show that if the C-k-smooth points of X are not dense in X for some k >= 1, then (R,<,+,X) interprets the monadic second-order theory of (N,+1). The same conclusion holds if the packing dimension of X is strictly greater than the topological dimension of X and X has no affine points.