  We prove that if $P,\mathcal{L}$ are finite sets of $\delta$-separated points
and lines in $\mathbb{R}^{2}$, the number of $\delta$-incidences between $P$
and $\mathcal{L}$ is no larger than a constant times
$$|P|^{2/3}|\mathcal{L}|^{2/3} \cdot \delta^{-1/3}.$$ We apply the bound to
obtain the following variant of the Loomis-Whitney inequality in the Heisenberg
group: $$ |K| \lesssim |\pi_{x}(K)|^{2/3} \cdot |\pi_{y}(K)|^{2/3}, \qquad K
\subset \mathbb{H}. $$ Here $\pi_{x}$ and $\pi_{y}$ are the vertical
projections to the $xt$- and $yt$-planes, respectively, and $|\cdot|$ refers to
natural Haar measure on either $\mathbb{H}$, or one of the planes. Finally, as
a corollary of the Loomis-Whitney inequality, we deduce that $$ \|f\|_{4/3}
\lesssim \sqrt{\|Xf\| \|Yf\| }, \qquad f \in BV(\mathbb{H}), $$ where $X,Y$ are
the standard horizontal vector fields in $\mathbb{H}$. This is a sharper
version of the classical geometric Sobolev inequality $\|f\|_{4/3} \lesssim
\|\nabla_{\mathbb{H}}f\|$ for $f \in BV(\mathbb{H})$.
