A Hausdorff–packing criterion for self-pinned distance sets
For a Borel subset E of the Euclidean plane with Hausdorff dimension d in (1, 5/4], a strict upper bound on its packing dimension by the explicit piecewise curve B_H(d) ensures a point y in E whose pinned distance set has positive one-dimensional Lebesgue measure. The three branches are 2d-1, 1+r_2(d-1), and 1/(3-2d), with the transitions and algebraic root defined independently in Challenge.lean. This is a sufficient criterion with an additional packing-dimension hypothesis, not a resolution of the general planar Falconer conjecture.
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