  Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two
different results pertaining to the \emph{tolerant testing} of juntas. Given
black-box access to a Boolean function $f:\{\pm1\}^{n} \to \{\pm1\}$, we give a
$poly(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between
functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far
from $k'$-juntas, where $k' = O(\frac{k}{\varepsilon^2})$.
  In the non-relaxed setting, we extend our ideas to give a
$2^{\tilde{O}(\sqrt{k/\varepsilon})}$ (adaptive) query algorithm that
distinguishes between functions that are $\gamma$-close to $k$-juntas and
$(\gamma+\varepsilon)$-far from $k$-juntas. To the best of our knowledge, this
is the first subexponential-in-$k$ query algorithm for approximating the
distance of $f$ to being a $k$-junta (previous results of Blais, Canonne, Eden,
Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required
exponentially many queries in $k$).
  Our techniques are Fourier analytical and make use of the notion of
"normalized influences" that was introduced by Talagrand [AoP, 1994].
