  Following \cite{Visintin}, we exploit the fractional perimeter of a set to
give a definition of fractal dimension for its measure theoretic boundary. We
calculate the fractal dimension of sets which can be defined in a recursive way
and we give some examples of this kind of sets, explaining how to construct
them starting from well known self-similar fractals. In particular, we show
that in the case of the von Koch snowflake $S\subset\mathbb R^2$ this fractal
dimension coincides with the Minkowski dimension, namely \begin{equation*}
P_s(S)<\infty\qquad\Longleftrightarrow\qquad
s\in\Big(0,2-\frac{\log4}{\log3}\Big). \end{equation*} We also study the
asymptotics as $s\to1^-$ of the fractional perimeter of a set having finite
(classical) perimeter.
