  Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over
$\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short
intervals when the $y$-coordinates lie in a prescribed interval, and the
distribution of the distance between consecutive $x$-coordinates with the same
property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We
study the distribution of the above distances with an extra condition that
$g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points
$P_i,P_{i+1}$.
