m at h . C A ] 9 M ay 2 00 3 FALCONER CONJECTURE IN THE PLANE FOR RANDOM METRICS
نویسندگان
چکیده
The Falconer conjecture says that if a compact planar set has Hausdorff dimension > 1, then the Euclidean distance set ∆(E) = {|x − y| : x, y ∈ E} has positive Lebesgue measure. In this paper we prove, under the same assumptions, that for almost every ellipse K, ∆ K (E) = {||x − y|| K : x, y ∈ E} has positive Lebesgue measure, where || · || K is the norm induced by an ellipse K. Equivalently, we prove that if a compact planar set has Hausdorff dimension > 1, then ∆(T E) has positive Lebesgue measure for almost every transformations T with bounded positive eigenvalues. We also use this result to deduce a version of the Erdos Distance Conjecture in the plane.
منابع مشابه
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تاریخ انتشار 2002