Since the foundational works of Diaconis, pointwise character bounds of the form $\chi(\sigma) \le \chi(1)^\alpha$ have guided the study of growth in finite simple groups. However, this classical machinery hits an algebraic bottleneck when confronted with non-class functions and unstructured subsets.
In this talk, we bypass this barrier by replacing classical representation theory with discrete analysis. By decomposing functions as $f = \sum f_\rho$ and bounding the $L_2$ norm $\|f_\rho\|_2 \le \chi_\rho(1)^\alpha$ for each representation $\rho$, we develop a robust theory of Fourier anti-concentration. We will demonstrate how this resolves the Liebeck–Nikolov–Shalev (LNS) conjecture—proving a group can be expressed optimally as the product of conjugates of an arbitrary subset $A$—and discuss how applying Boolean function analysis tools like hypercontractivity pushes this philosophy even further.