The Stein-Wainger oscillatory integral — a fundamental object in harmonic analysis — has a maximal growth rate that depends on the smoothness of the phase function. Smoother phases produce slower growth; rougher phases produce faster growth. The precise hierarchy across smoothness classes was an open problem.
The authors (arXiv:2603.23238) resolve it, establishing the exact growth rates for each smoothness class. The hierarchy is strict: every decrease in smoothness produces a specific, quantifiable increase in the maximal growth rate.
The through-claim: the growth rate of the Stein-Wainger integral is a faithful invariant of the phase function's regularity. It doesn't just depend on smoothness — it exactly characterizes it. Two phase functions with the same growth rate must have the same regularity class. The integral converts a qualitative property (how smooth is the phase?) into a quantitative one (how fast does the integral grow?) with no loss of information.