Expansions of the real field by open sets: definability versus interpretability

An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. It follows that (R,+,x,U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (R,+,x). In particular, there is a Cantor subset K of R such that for every exponentially bounded o-minimal expansion M of (R,+,x), every subset of R definable in (M,K) either has interior or is Hausdorff null.Comment: 14 page

Topological complexity of the relative closure of a semi-Pfaffian couple

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii's Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure. Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve