Infinite combinatorics and the foundations of regular variation
The `uniform convergence theorem' is a statement that the pointwise convergence \(h(x+t)-h(x)\to 0\) for \(x\to\infty\) (slowly varying \(h:\mathbb R\to\mathbb R\) in the additive formulation) implies the locally uniform convergence. The validity of this statement is shown to be equivalent to a combinatorial principle saying that for every \(\epsilon>0\), \(x_n\to\infty\), and bounded sequence \((u_n)\), some shift of \(\bigcap_{n=k}^\infty \{t:|h(t+x_n)-h(x_n)|<\epsilon\}\) for some \(k\) contains a subsequence of \((u_n)\). The combinatorial principle was extracted from a close reading of the standard treatment of uniform convergence theorem in the monograph by [\textit{N. H. Bingham, C. M. Goldi} and \textit{J. L. Teugels}, Regular Variation. Encyclopedia of Mathematics and its applications, Vol. 27. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.26001)], as the authors say. It covers and unifies the cases of functions having the Baire property and Lebesgue measurable functions, when the uniform convergence statement holds. This solves a problem posed in the above mentioned monograph. Some other combinatorial principles are also shown to be equivalent, e.g., \(\lim_{n\to\infty} h(t_n+x_n)-h(x_n)=0\) for \((t_n)\) bounded and \(x_n\to \infty\).
scientific article; zbMATH DE number 5626928
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