  Jorgensen and Pedersen have proven that a certain fractal measure $\nu$ has
no infinite set of complex exponentials which form an orthonormal set in
$L^2(\nu)$. We prove that any fractal measure $\mu$ obtained from an affine
iterated function system possesses a sequence of complex exponentials which
forms a Riesz basic sequence, or more generally a Bessel sequence, in
$L^2(\mu)$ such that the frequencies have positive Beurling dimension.
