CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2008-21

September 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) =λρ 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 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) ≤ 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.

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