Junta distance approximation with sub-exponential queries

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f : {±1}n → {±1}: 1. We give a [EQUATION] query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ + ε)-far from k'-juntas, where [EQUATION]. 2. In the non-relaxed setting, we extend our ideas to give a [EQUATION] (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ + ε)-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 [32].