How Close is your Function to Depending on a Small Number of its Inputs?

This thesis focuses on the following question: if we have query access to some f : {−1, 1} → {−1, 1} (meaning we can query any of its 2 outputs at will), how many queries do we need to discern whether f is “close” or “far” from depending on only a subset of k of its inputs? Such functions are known in the field as “k-juntas”. The main results come from the recent work of [ITW21], which gave an improved upper bound on the query complexity of this problem. Namely, it was shown that only 2 √ k) queries to f suffice to answer this question. Along the way, we will provide a more detailed and leisurely survey of the other results and progress in this area, examining other techniques and the strongest known upper and lower bounds for variants of this problem.