Restricted families of projections onto planes: the general case of nonvanishing geodesic curvature (Q6058047)

Summary: It is shown that if \(\gamma : [a, b] \to S^2\) is \(C^3\) with \(\det(\gamma, \gamma', \gamma'') \neq 0\), and if \(A \subseteq \mathbb{R}^3\) is a Borel set, then \(\dim \pi_{\theta} (A) \geq \min \{2, \dim A, \dim A/2 + 3/4\}\) for a.e. \(\theta \in [a, b]\), where \(\pi_{\theta}\) denotes projection onto the orthogonal complement of \(\gamma(\theta)\) and ``dim'' refers to Hausdorff dimension. This partially resolves a conjecture of \textit{K. Fässler} and \textit{T. Orponen} [Proc. Lond. Math. Soc. (3) 109, No. 2, 353--381 (2014; Zbl 1302.28015)] in the range \(1 < \dim A \leq 3/2\), 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.