Mattila and Sjolin (1999) proved that if a subset of \(\mathbb{R}^d\) has Hausdorff measure greater than \((d+1)/2\), then its distance set has nonempty interior in \(\mathbb{R}\). I will describe how to generalize this result to a wide range of \(k\)-point configuration sets, using microlocal analysis and an optimization procedure. This is joint work with Alex Iosevich and Krystal Taylor.