We used photometric data from the WASP (Wide-Angle Search for Planets) survey to explore the possibility of detecting eclipses and transit signals of brown dwarfs, gas giants and terrestrial companions in close orbit around white dwarfs. We performed extensive Monte Carlo simulations and we found that for Gaussian random noise WASP is sensitive to companions as small as the Moon orbiting a $Vsim$12 white dwarf. For fainter stars WASP is sensitive to increasingly larger bodies. Our sensitivity drops in the presence of co-variant noise structure in the data, nevertheless Earth-size bodies remain readily detectable in relatively low S/N data. We searched for eclipses and transit signals in a sample of 194 white dwarfs in the WASP archive however, no evidence for companions was found. We used our results to place tentative upper limits to the frequency of such systems. While we can only place weak limits on the likely frequency of Earth-sized or smaller companions; brown dwarfs and gas giants (radius$simeq$ R$_{jup}$) with periods $leq$0.2 days must certainly be rare ($<10%$). More stringent constraints requires significantly larger white dwarf samples, higher observing cadence and continuous coverage. The short duration of eclipses and transits of white dwarfs compared to the cadence of WASP observations appears to be one of the main factors limiting the detection rate in a survey optimised for planetary transits of main sequence stars.
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