Expansions of o-minimal structures by sparse sets

Expansions of o-minimal structures by dense independent sets

Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that ⋃ H is definably independent in M. We study the structure (M, (H)H∈H). Positive results include that every open set definable in (M, (H)H∈H) is definable in M, the structure induced in (M, (H)H∈H) on any H0 ∈ H is as simple as possible (in a sense that is made preci...

Expansions of o-minimal structures by fast sequences

Topologizing interpretable sets in O-minimal Structures

Let M be a structure in some language. Assume M has elimination of imaginaries. Let X be a definable set. Definable will mean “definable with parameters.” By a definable topology, we mean a definable family of subsets {By ⊂ X}y∈Y which form the basis for some topology on X. The fact that these form a basis for a topology amounts to the claim that if y1, y2 have By1 ∩By2 6= ∅, then for every x ∈...

Near Integral Points of Sets Definable in O Minimal Structures

Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly exponentially near to many integral points, then there is an unbounded set, which is R definable without parameters, and which is exponentially near to S.

Bifurcation Sets of Definable Functions in O-minimal Structures

In this work we answer a question stated by Loi and Zaharia concerning trivialization of definable functions off the bifurcation set: we prove that definable functions are trivial off the bifurcation set, and the trivialization can be chosen definable.