Patrick Speissegger : A Borel Lemma for o-minimal structures

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

April 11, 2006, 4 p.m.

In

SEO 427

Abstract

Let f be a totally defined real-valued function on the real line. Borel's Lemma states that if f is increasing and everywhere greater than or equal to 1, and if r>1 is fixed, then the set {x: f(x + 1/f(x)) >= r f(x)} has outer measure at most r/(r-1). Recently, Chris Miller conjectured that a suitably restated version of Borel's Lemma was true in the o-minimal setting. Interestingly, the proof of this version appears to be more elementary for exponential o-minimal structures than for power-bounded ones. (Joint work with Alf Dolich and Chris Miller)