  Nils Tongring (1987) proved sufficient conditions for a compact set to
contain $k$-tuple points of a Brownian motion. In this paper, we extend these
findings to the fractional Brownian motion. Using the property of strong local
nondeterminism, we show that if $B$ is a fractional Brownian motion in
$\mathbb{R}^d$ with Hurst index $H$ such that $Hd=1$, and $E$ is a fixed,
nonempty compact set in $\mathbb{R}^d$ with positive capacity with respect to
the function $\phi(s) = (\log_+(1/s))^k$, then $E$ contains $k$-tuple points
with positive probability. For the $Hd > 1$ case, the same result holds with
the function replaced by $\phi(s) = s^{-k(d-1/H)}$.
