Infinite intersections of doubling measures, weights, and function classes

A series of long-standing questions in harmonic analysis ask whether the intersection of all prime “ p -adic versions” of an object, such as a doubling measure or a Muckenhoupt or reverse Hölder weight, recovers the full object. Investigation into these questions was reinvigorated in 2019 by work of Boylan–Mills–Ward, culminating in showing that this recovery fails for a finite intersection in work of Anderson–Bellah–Markman–Pollard–Zeitlin. Via generalizing a new number theoretic construction therein, we answer these questions.