  The emergence of a compressible insulator phase, known as the Bose glass, is
characteristic of the interplay of interactions and disorder in correlated Bose
fluids. While widely studied in tight-binding models, its observation remains
elusive owing to stringent temperature effects. Here we show that this issue
may be overcome by using Lieb-Liniger bosons in shallow quasiperiodic
potentials. A Bose glass, surrounded by superfluid and Mott phases, is found
above a critical potential and for finite interactions. At finite temperature,
we show that the melting of the Mott lobes is characteristic of a fractal
structure and find that the Bose glass is robust against thermal fluctuations
up to temperatures accessible in quantum gases. Our results raise questions
about the universality of the Bose glass transition in such shallow
quasiperiodic potentials.
