  We study divergence properties of Fourier series on Cantor-type fractal
measures, also called mock Fourier series. We show that in some cases the
$L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and
therefore the Fourier series are not even pointwise convergent. We apply these
results to the Lebesgue measure to show that a certain rearrangement of the
exponential functions, which we call scrambled Fourier series, have a
corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which
is much worse than the known logarithmic bound. The divergence properties are
related to the Mahler measure of certain polynomials and to spectral properties
of Ruelle operators.
