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].
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