A WEAK GROWTH DICHOTOMY FOR D-MINIMAL EXPANSIONS OF THE REAL FIELD*

CHRIS MILLER

Version: December 15, 2006

An expansion ℜ of the real line is d-minimal if for every n ∈ ℕ and definable A ⊆ ℝ^{n+1}, there exists N ∈ ℕ such that for every x ∈ ℝ^{m}, the fiber { t ∈ ℝ : (x, t) ∈ A } either has interior or is a union of N (not necessarily distinct) discrete sets. For more information on d-minimality, see [1, 2, 3, 5, 6, 8].

Until further notice, ℜ denotes an expansion of the real field; definability is with respect to ℜ. We obtain a generalization of a result that holds for o-minimal expansions of \overline{ℝ}.

1. Theorem. If ℜ is d-minimal, then either ℜ is exponential or every definable power function is ∅-definable.

The above is not quite a dichotomy—every power function is ∅-definable in the o-min-imal (hence d-minimal) structure ( ℝ, +, e^{x}, (r)_{r∈ℝ} )—but having dcl(∅) = ℝ is the only obstruction to a dichotomy.

The property of all definable power functions being ∅-definable has been quite useful in o-minimality; perhaps the same will be true for d-minimality (but it’s probably too early to tell).

By the usual tricks, the theorem is an easy consequence of the next four results.

2. Lemma. Let F: ℝ^{m+1} → ℝ be ∅-definable. Then { (a, r) ∈ ℝ^{m+1} : F (a, ·) = x^{r} } is ∅-definable.

Proof. For all (a, r) ∈ ℝ^{m+1}, we have F (a, ·) = x^{r} if and only if F (a, 1) = 1, F (a, ·)↾(0, ∞) is differentiable, and t(∂F/∂t)(a, t) = rF (a, t) for all t > 0. □

3. Lemma. If ℜ has definable Skolem functions and there exist m ∈ ℕ and definable F: ℝ^{m+1} → ℝ such that { r ∈ ℝ : ∃a ∈ ℝ^{m}, F (a, ·) = x^{r} } has interior, then ℜ is exponential.

Proof. Argue as in [4, 4.1], using the previous lemma. □

4. Proposition ([3]). If ℜ is d-minimal, then ℜ has definable Skolem functions.

5. Lemma. Suppose that every ∅-definable subset of ℝ either has interior or is a finite union of discrete sets.

(1) If A ⊆ ℝ is ∅-definable and has no interior, then A ⊆ dcl(∅).

(2) Every nonempty ∅-definable set (of any arity) contains a ∅-definable point.

Proof. (1). Since dcl(∅) is dense in ℝ, every nonempty open interval intersects dcl(∅), and every isolated point of a ∅-definable subset of ℝ is ∅-definable. The set of isolated points of a ∅-definable set is ∅-definable.

(2) follows from (1) by an easy induction. □

*This is not a preprint; please do not refer to it as such.

By similar arguments:

6. Theorem. If ℜ is a d-minimal expansion of (ℝ, <, +, 1), then either ℜ defines multiplication or every definable scalar function is ∅-definable.

Details are left to the interested reader; see e.g. [7] for some relevant tricks.

References

1. H. Friedman and C. Miller, Expansions of o-minimal structures by fast sequences, J. Symbolic Logic 70 (2005), 410–418.

2. C. Miller, D-minimal expansions of the real field have the exchange property, unpublished note, available on Miller’s web page, see below for URL.

3. , Definable choice in d-minimal expansions of ordered groups, unpublished note, available on Miller’s web page, see below for URL.

4. , A growth dichotomy for o-minimal expansions of ordered fields, Logic: From Foundations to Applications, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, pp. 385–399.

5. , Tameness in expansions of the real field, Logic Colloquium ’01 (Vienna, 2001), Lect. Notes in Logic, vol. 20, Assoc. Symbolic Logic, Urbana, IL, 2005, pp. 281–316.

6. , Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (2006), 1483–1493.

7. C. Miller and S. Starchenko, A growth dichotomy for o-minimal expansions of ordered groups, Trans. Amer. Math. Soc. 350 (1998), 3505–3521.

8. C. Miller and J. Tyne, Expansions of o-minimal structures by iteration sequences, Notre Dame J. Formal Logic 47 (2006), 93–99.

Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210, USA

  E-mail address: miller@math.ohio-state.edu
  URL: http://www.math.ohio-state.edu/~miller