{"id":"palomar/PALOMAR-2026-10-06-000008","url":"https://github.com/CoolRmal/falconer-packing/tree/70140ccedfb6de71342299523a21b1550df69ab9","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-06-000008-v1.json","repo":"https://github.com/CoolRmal/falconer-packing","tags":["math.CA","msc:28A80","msc:42B10","geometric-measure-theory","distance-sets","hausdorff-dimension","falconer-conjecture"],"score":0,"title":"A Hausdorff–packing criterion for self-pinned distance sets","trust":"high","domain":"math.CA","source":"palomar","authors":["Yongxi Lin"],"reviews":0,"subject":"math.CA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-06","reactions":{},"formalized":true,"formalizes":[{"title":"A packing-dimension condition for positive pinned distances","identifier":"falconer-packing-theorem, 30 September 2026, Theorem 1.1","relationship":"formalizes"}]}