  For a family of fractal measures, we find an explicit Fourier duality. The
measures in the pair have compact support in $\br^d$, and they both have the
same matrix scaling. But the two use different translation vectors, one by a
subset $B$ in $\br^d$, and the other by a related subset $L$. Among other
things, we show that there is then a pair of infinite discrete sets $\Gamma(L)$
and $\Gamma(B)$ in $\br^d$ such that the $\Gamma(L)$-Fourier exponentials are
orthogonal in $L^2(\mu_B)$, and the $\Gamma(B)$-Fourier exponentials are
orthogonal in $L^2(\mu_L)$. These sets of orthogonal "frequencies" are
typically lacunary, and they will be obtained by scaling in the large. The
nature of our duality is explored below both in higher dimensions and for
examples on the real line.
  Our duality pairs do not always yield orthonormal Fourier bases in the
respective $L^2(\mu)$-Hilbert spaces, but depending on the geometry of certain
finite orbits, we show that they do in some cases. We further show that there
are new and surprising scaling symmetries of relevance for the ergodic theory
of these affine fractal measures.
