CDAM: Computational, Discrete and Applicable Mathematics@LSE 

CDAM Research Report, LSECDAM200821September 2008 
Regular variation without limits
N. H. Bingham and A. J. Ostaszewski
Karamata theory ([BGT] Ch. 1) explores functions f for which the limit function g(λ) := f(λx)/f(x) exists (as x → ∞) and for which g(λ) =λ^{ρ} subject to mild regularity assumptions on f. Further Karamata theory ([BGT] Ch. 2) explores functions f for which the upper limit f*(λ) := lim sup f(λx)/f(x), as x → ∞, remains bounded. Here the usual regularity assumptions invoke boundedness of f* on a Baire nonmeagre/measurable nonnull set, with f Baire/measurable, and the conclusions assert uniformity over compact λsets (implying upper bounds of the form f(λx)/f(x) ≤ K λ^{&rho} 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.A PDF file (248 kB) with the full contents of this report can be downloaded by clicking here.
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