Yet Another Proof of Marstrand’s Theorem
نویسندگان
چکیده
through the origin containing vθ and projθ : R 2 → Lθ the orthogonal projection. From now on, we’ll restrict θ to the interval [−π/2, π/2], because Lθ = Lθ+π. In 1954, J. M. Marstrand [10] proved the following result on the fractal dimension of plane sets. Theorem 1.1. If K ⊂ R is a Borel set such that HD(K) > 1, then m(projθ(K)) > 0 for m-almost every θ ∈ R. The proof is based on a qualitative characterization of the “bad” angles θ for which the result is not true. Specifically, Marstrand exhibits a Borel measurable function f(x, θ), (x, θ) ∈ R × [−π/2, π/2], such that f(x, θ) = ∞ for ms-almost every x ∈ K, for every “bad” angle. In particular,
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