Regular variation without limits

Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 1], ) explores functions f for which the limit function g(λ):=f(λx)/f(x) exists (as x→∞) and for which g(λ)=λρ subject to mild regularity assumptions on f. Further Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 2]) explores functions f for which the upper limit , as x→∞, remains bounded. Here the usual regularity assumptions invoke boundedness of f* on a Baire non-meagre/measurable non-null set, with f Baire/measurable, and the conclusions assert uniformity over compact λ-sets (implying upper bounds of the form f(λx)/f(x)⩽Kλρ for all large λ, x). We give unifying combinatorial conditions which include the two classical cases, deriving them from a combinatorial semigroup theorem. We examine character degradation in the passage from f to f* (using some standard descriptive set theory) and thus identify natural classes in which the theory may be established.