Abstract
According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝn has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject.
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References
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dimA >n/2 would be the optimal result forn ≥ 2.
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Bourgain, J. Hausdorff dimension and distance sets. Israel J. Math. 87, 193–201 (1994). https://doi.org/10.1007/BF02772994
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DOI: https://doi.org/10.1007/BF02772994