Bessel sequences of exponentials on fractal measures

Abstract

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.