Centre for Discrete and Applicable Mathematics

CDAM Research Report, LSE-CDAM-2006-22

January 2007 (Revised November 2008)

Infinite Combinatorics and the foundations of regular variation

N. H. Bingham and A. J. Ostaszewski

In memoriam Paul Erdös, 1913-1996

The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have common generalizations, exemplified by `containment up to translation of subsequences'. All of our combinatorial regularity properties are equivalent to the uniform convergence property.
Keywords: Regular variation, uniform convergence theorem, Cauchy functional equation, Baire property, measurability, density topology, measure-category duality, infinite combinatorics, subuniversal set, No Trumps Principle.

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