A major question in the study of black hole binaries involves our understanding of the accretion geometry when the sources are in the hard state. In this state, the X-ray energy spectrum is dominated by a hard power-law component and radio observations indicate the presence of a steady and powerful compact jet. Although the common hard state picture is that the accretion disk is truncated, perhaps at hundreds of gravitational radii from the black hole, recent results for the recurrent transient GX 339-4 by Miller and co-workers show evidence for optically thick material very close to the black holes innermost stable circular orbit. That work focused on an observation of GX 339-4 at a luminosity of about 5% of the Eddington limit and used parameters from a relativistic reflection model and the presence of a soft, thermal component as diagnostics. In this work, we use similar diagnostics, but extend the study to lower luminosities (2.3% and 0.8% Ledd) using Swift and RXTE observations of GX 339-4. We detect a thermal component with an inner disk temperature of ~0.2 keV at 2.3% Ledd. We detect broad features due to iron Kalpha that are likely related to reflection of hard X-rays off the optically thick material. If these features are broadened by relativistic effects, they indicate that optically thick material resides within 10 Rg down to 0.8% Ledd, and the measurements are consistent with the inner radius of the disk remaining at ~4 Rg down to this level. However, we also discuss an alternative model for the broadening, and we note that the evolution of the thermal component is not entirely consistent with the constant inner radius interpretation. Finally, we discuss the results in terms of recent theoretical work on the possibility that material may condense to maintain an inner optically thick disk.
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