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)
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