Improving Kaufman's exceptional set estimate for packing dimension

Abstract

Given $0 < s < 1$, I prove that there exists a constant $\epsilon = \epsilon(s) > 0$ such that the following holds. Let $K \subset \mathbb{R}^{2}$ be a Borel set with $\mathcal{H}^{1}(K) > 0$, and let $E_{s}(K) \subset S^{1}$ be the collection of unit vectors $e$ such that $$\dim_{\mathrm{p}} \pi_{e}(K) \leq s.$$ Then $\dim_{\mathrm{H}} E_{s}(K) \leq s - \epsilon$.