To wrap up the notes on o-minimal structures, let’s consider two questions that were still open at the end of the last millennium:
Questions
- Does every o-minimal expansion of the real field admit analytic cell decomposition?
- Is there a unique maximal o-minimal expansion of the real field?
The answer to both questions is “no”, and I want to briefly summarize here how this follows from the theorems we proved here and from some other theorems known before the turn of the millenium but not proved here.
Analytic cell decomposition
Replacing the term “$C^p$” in the definitions of cell and cell decomposition by “$C^\infty$” or “analytic”, we obtain the definitions of $C^\infty$-cell and $C^\infty$-cell decomposition or analytic cell and analytic cell decomposition, respectively.
Fact
All examples of o-minimal expansions of the real field known by the year 2000 admit analytic cell decomposition.
These examples include the structure $\Ran$ discussed here, as well as several expansions of $\Ran$ found here and here. Moreover, if $\R$ is an o-minimal expansion of the real field that admits analytic cell decomposition, then so does its pfaffian closure $\P(\R)$ discussed here.
On the other hand, the proof of the $C^p$-cell decomposition theorem works only for finite $p$, because the concepts of being $C^\infty$ or analytic are not first-order definable, as Wilkie succinctly shows here for the $C^\infty$ case.
Maximal o-minimal expansions
An o-minimal expansion $\R$ of the real field is maximal, if every proper expansion of $\R$ is not o-minimal.
By Zorn’s Lemma, every o-minimal expansion of the real field has a maximal o-minimal expansion. But is there a unique maximal o-minimal expansion of the real field, that is, are all these maximal ones identical?
This question is related to the question whether two given o-minimal expansions $\R_1$ and $\R_2$ of the real field can be amalgamated, that is, whether the structure generated by all sets definable in $\R_1$ or $\R_2$ is again o-minimal.
For example, the expansion of the real field by the restriction of the Euler Gamma function to $(0,\infty)$ is o-minimal, as is the expansion of the real field by the restriction of the Riemann zeta function to $(1,\infty)$ (see p. 515 of this paper); but it is not known whether the expansion of the real field by both these restrictions is o-minimal. (I conjecture that it is.)
Back to the questions
Both questions were first answered by finding suitable examples of quasianalytic classes:
Theorem (see Theorem 2 here)
- There exists a quasianalytic class $\C$ satisfying Axioms (C1)–(C4) such that $\C_1$ contains a germ that is nowhere analytic.
- Given any $\C^\infty$-function $f:[-1,1] \into \RR$, there exist quasianalytic classes $\C$ and $\D$, each satisfying Axioms (C1)–(C4), and there exist $g \in \C_1$ and $h \in \D_1$ such that $f = g+h$.
Part 1 of this theorem answers Question 1, since the corresponding o-minimal structure $\RR_\C$ contains a definable function that is nowhere analytic; hence the graph of this function cannot be a finite disjoint union of analytic cells.
Part 2 of this theorem answers Question 2, since both $\RR_\C$ and $\RR_\D$ are o-minimal, while the function $f$ is definable in any expansion of the real field that expands both $\RR_\C$ and $\RR_\D$. Thus, we can take $f:[-1,1] \into \RR$ defined by $$f(x):= \begin{cases} e^{-1/x} \sin\left(\frac1x\right) &\text{if } x > 0, \\ 0 &\text{if } x \le 0 \end{cases}$$ to obtain two examples of non-amalgamable o-minimal structures.
Final remarks
Question 1 also has “no” as an answer when “analytic” is replaced by “$C^\infty$”, as a clever genericity construction by Le Gal and Rolin shows.
A much more intriguing answer to Question 2 is given here by Rolin, Sanz and Schäfke: they show that there exists an analytic 1-distribution $d$ on $\RR^3$, definable in the real field, such that every single integral manifold of $d$ generates an o-minimal expansion of the real field, while the expansion of the real field by any two distinct integral manifolds of $d$ is not o-minimal!