A lower bound for the a.e. behaviour of Hausdorff dimension under vertical projections in the Heisenberg group

Terence L. J. Harris

Published 2018-11-30Version 1

It is shown that for a Borel set $A$ in the Heisenberg group with $\dim A >2$, \[ \dim P_{\mathbb{V}^{\perp}_{\theta}} A \geq \begin{cases} \frac{\dim A}{2} &\text{ if } \dim A \in \left(2, \frac{5}{2} \right] \\ \frac{ \dim A(\dim A + 2)}{4\dim A -1} &\text{ if } \dim A \in \left( \frac{5}{2} , 4 \right], \end{cases} \] for a.e. $\theta \in [0,\pi)$, where $\dim$ refers to the Hausdorff dimension under the Kor\'anyi metric, and $P_{\mathbb{V}^{\perp}_{\theta}}$ is the vertical Heisenberg projection onto the vertical plane at angle $\theta+ \frac{\pi}{2}$. This improves the known lower bounds in the range $2 < \dim A <\frac{12+\sqrt{109}}{7}$.