Restricted families of projections onto planes: The general case of nonvanishing geodesic curvature

Abstract

. It is shown that if γ : [ a,b ] → S 2 is C 3 with det( γ,γ ′ ,γ ′′ ) 6 = 0, and if A ⊆ R 3 is a Borel set, then dim π θ ( A ) ≥ min n 2 , dim A, dim A 2 + 34 o for a.e. θ ∈ [ a,b ], where π θ denotes projection onto the orthogonal complement of γ ( θ ) and “dim” refers to Hausdorff dimension. This partially resolves a conjecture of F¨assler and Orponen in the range 1 < dim A ≤ 3 / 2, which was previously known only for non-great circles. For 3 / 2 < dim A < 5 / 2 this improves the known lower bound for this problem.