  We present X-ray and radio observations of what may be the closest type Iax
supernova (SN) to date, SN 2014dt (d=12.3-19.3 Mpc) and provide tight
constraints on the radio and X-ray emission. We infer a specific radio
luminosity of < (1.0-2.4)E25 erg/s/Hz at a frequency of 7.5 GHz and a X-ray
luminosity < 1.4E38 erg/s (0.3-10 keV) at ~38-48 days post-explosion. We
interpret these limits in the context of Inverse Compton (IC) emission and
synchrotron emission from a population of electrons accelerated at the forward
shock of the explosion in a power-law distribution $N_e(\gamma_e)\propto
\gamma_e^{-p}$ with p=3. Our analysis constrains the progenitor system
mass-loss rate to be smaller than 5E-6 solar masses per year at distances where
r <= 1E16 cm for an assumed wind velocity v=100 km/s, and a fraction of
post-shock energy into magnetic fields and relativistic electrons of
epsilon_B=0.01 and epsilon_e=0.1, respectively. This result rules out some of
the parameter space of symbiotic giant star companions, and it is consistent
with the low mass-loss rates expected from He-star companions. Our calculations
also show that the improved sensitivity of the next generation Very Large Array
(ngVLA) is needed to probe the very low-density media characteristic of He
stars that are the leading model for binary stellar companions of white dwarfs
giving origin to type Iax SNe.
