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 $${\dim }_{\mathrm{p}}{\pi }_e(K)\le s.$$ Then ${\dim }_{\mathrm{H}}{E}_s(K)\le s-\epsilon$.

22 pages. v2: Significantly improved main result; changed title accordingly