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 ( λ x ) / f ( x ) exists (as x → ∞ ) and for which g ( λ ) = λ ρ 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 f * ( λ ) : = lim sup f ( λ x ) / f ( x ) , 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 ( λ x ) / f ( x ) ⩽ K λ ρ 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.