A ROSAT deep survey of four small gamma-ray burst error boxes

We have used the ROSAT High Resolution Imager to search for quiescent X-ray counterparts to four gamma-ray bursts (GRBs) which were localized to small (less than or equal to 10 arcmin(2)) error boxes with the Interplanetary N etwork. The observations took: place years after the bursts, and the effect ive exposure times for each target varied from similar to 16 to 23 ks. We h ave not found any X-ray sources inside any of the error boxes. The 0.1-2.4 keV 3 sigma flux upper limits range from around 5 x 10(-14) to 6 x 10(-13) ergs cm(-2) s(-1) depending on the burst and the assumed shape of the quies cent spectrum. Pie consider four types of X-ray-emitting galaxies (normal, active galactic nucleus, faint, and star forming) and use the flux upper li mits to constrain their redshifts. We then use the GRB fluences to constrai n the total energies of the bursts.