Submission date: 10 January 2019

Abstract:

Let X ⊆ ℝ^n be closed. If the C^k-smooth points of X are not dense in X for some k ≥ 0, then (ℝ,<,+,0,X) interprets the monadic second order theory of (ℕ,+1). The same conclusion holds if the Hausdorff dimension of X is strictly greater than the topological dimension of X and X has no affine points. Thus, if X is virtually any fractal subset of ℝ^n, then (ℝ,<,+,0,X) interprets the monadic second order theory of (ℕ,+1).

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Full text arXiv 1901.03273: pdf, ps.