  A rapid timing analysis of VLT/ULTRACAM and RXTE observations of the black
hole binary GX 339-4 in its 2007 low/hard state is presented. The optical light
curves in the r', g' and u' filters show slow (~20 s) quasi-periodic
variability. Upon this is superposed fast flaring activity on times approaching
the best time resolution probed (~50 ms) and with maximum strengths of more
than twice the local mean. Power spectral analysis over ~0.004-10 Hz is
presented, and shows that although the average optical variability amplitude is
lower than that in X-rays, the peak variability power emerges at a higher
Fourier frequency in the optical. Energetically, we measure a large optical vs.
X-ray flux ratio, higher than that seen when the source was fully
jet-dominated. Such a large ratio cannot be easily explained with a disc alone.
The optical:X-ray cross-spectrum shows a markedly different behaviour above and
below ~0.2 Hz. The peak of the coherence function above this threshold is
associated with a short optical time lag, also seen as the dominant feature in
the time-domain cross-correlation at ~150 ms. The rms energy spectrum of these
fast variations is best described by distinct physical components over the
optical and X-ray regimes, and also suggests a maximal disc fraction of 20% at
~5000 A. If the constant time delay is due to propagation of fluctuations to
(or within) the jet, this is the clearest optical evidence to date of the
location of this component. The low-frequency QPO is seen in the optical but
not in X-rays. Evidence of reprocessing emerges at the lowest Fourier
frequencies, with optical lags at ~10 s and strong coherence in the blue u'
filter. Simultaneous optical spectroscopy also shows the Bowen fluorescence
blend, though its emission location is unclear. But canonical disc reprocessing
cannot dominate the optical power easily, nor explain the fast variability.
(abridged)
