  Recent experimental reports suggested the existence of a finite-temperature
insulator in the vicinity of the superconductor-insulator transition. The rapid
decay of conductivity over a narrow temperature range was theoretically linked
to both a finite-temperature transition to a many-body-localized state, and to
a charge-Berezinskii Kosterlitz Thouless transition. Here we report of
low-frequency noise measurements of such insulators to test for many body
localization. We observed a huge enhancement of the low-temperatures noise when
exceeding a threshold voltage for nonlinear conductivity and discuss our
results in light of the theoretical models.
