EXPANSIONS OF THE REAL FIELD BY OPEN SETS: DEFINABILITY VERSUS INTERPRETABILITY

HARVEY FRIEDMAN, KRZYSZTOF KURDYKA, CHRIS MILLER, AND PATRICK SPEISSEGGER

Abstract. An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, 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, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, E) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion ℜ of (ℝ, +, ·), every subset of ℝ definable in (ℜ, E) either has interior or is Hausdorff null.

The reader is assumed to be familiar with the basics of first-order definability over the real field \overline{ℝ} := (ℝ, +, ·), especially o-minimality. Requisite material can be found in van den Dries and Miller [4]. We refer to Kechris [9] and Mattila [11] for basic descriptive set theory and geometric measure theory. We say that a subset of ℝ^{n} is constructible if it is a boolean combination of open subsets of ℝ^{n}. By Dougherty and Miller [1], every constructible E ⊆ ℝ^{n} is a boolean combination of open sets that are definable in (ℝ, <, E); we use this fact without further mention.

We begin with a simply-stated question: What can be said about the sets that are definable, allowing arbitrary real parameters, in an expansion ℜ of \overline{ℝ} by a collection of constructible subsets of ℝ? First, every quantifier-free definable set is constructible, hence Borel. Next, every existentially definable set is Σ_{1}^{1} (also known as Souslin, Suslin, or analytic), and every universally definable set is Π_{1}^{1} (also known as co-analytic). Continuing in this fashion, every definable set is projective in the sense of descriptive set theory. All real projective sets are definable in (\overline{ℝ}, ℕ) [9, 37.6]. As a result, there are now many examples known where ℜ defines all real projective sets; see [8,13,14] for some non-obvious ones. On the other hand, there are now many examples known where every definable set is constructible; see [5,6,13,16] for some non-o-minimal ones. Heretofore, no other behaviors have been documented. We show in this paper that there is at least one other possibility: ℜ can define sets in every Σ_{n+1}^{1} \ Σ_{n}^{1}, and thus of every projective level, yet not define all real projective sets. As Borel isomorphisms preserve projective level, this is immediate from

Theorem A. There is a closed E ⊆ ℝ such that:

(a) (\overline{ℝ}, E) defines a Borel isomorph of (\overline{ℝ}, ℕ).

(b) Every unary (that is, contained in ℝ) set definable in (\overline{ℝ}, E) either has interior or is nowhere dense.

Expansions of \overline{ℝ} in which every unary definable set either has interior or is nowhere dense have a number of good properties [13], but we shall not dwell on this here. On the other hand, this condition is not strong enough to rule out a significant difference between interpretability and definability of interesting algebraic objects.

February 14, 2010. Some version of this document is to appear in J. Symbolic Logic.

Corollary. With E as in Theorem A, (\overline{ℝ}, E) defines no proper nontrivial subgroups of (ℝ, +), nor any proper noncyclic subgroups of (ℝ^{>0}, ·), yet (\overline{ℝ}, E) interprets every real projective set, in particular, every projective subfield of \overline{ℝ}.

As we shall see (Theorem B, below), we can choose E so that part (b) of Theorem A holds even for (\overline{ℝ}, exp, E), in which case neither does (\overline{ℝ}, E) define any proper nontrivial subgroups of (ℝ^{>0}, ·).

We postpone beginning the proof proper of Theorem A, but we outline some of the main ideas now. In order to satisfy part (b), it suffices by [5, Theorem A] and cell decomposition to produce a closed E ⊆ ℝ such that:

(b)′ For every n ∈ ℕ, bounded open semialgebraic cell U ⊆ ℝ^{n}, and bounded continuous semialgebraic f: U → ℝ, the image f(U ∩ E^{n}) is nowhere dense, where E^{n} is the n-th cartesian power of E.

Given any uncountable Σ_{1}^{1} set E ⊆ ℝ such that (b)′ holds, there is a Cantor set^{1} K such that both (a) and (b)′ hold with E replaced by K; this is fairly easy modulo known descriptive set theory and some definability tricks (1.14, below). Thus, it suffices to find a Cantor set E such that condition (b)′ holds for E. In order to motivate further developments, we describe a naive approach that we could not make work. Let E be a Cantor set such that every E^{n} is Hausdorff null (that is, has Hausdorff dimension zero). By cell decomposition, we reduce further to the case that f is C^{1} and nowhere locally constant. Write U as the union of the compact sets A_{r}, r > 0, where A_{r} is the set of x ∈ U whose distance to the boundary of U is at least 1/r. Each restriction f↾A_{r} is Lipshitz, so f(E^{n} ∩ A_{r}) is compact and Hausdorff null. Since f is nowhere locally constant, f(E^{n} ∩ A_{r}) is also Cantor for all sufficiently large r. Hence, f(U ∩ E^{n}) is the union of a “semialgebraically parameterized” increasing family of Hausdorff null Cantor sets. But we see no way to conclude from this that f(U ∩ E^{n}) is nowhere dense, which is required in order to employ the aforementioned technology from [5]. The fundamental shortcoming of this approach is that it appears not to account for how limit points of f(U ∩ E^{n}) are formed at the boundary of U. We overcome this by a more careful choice of E based on an analysis of the behavior of bounded semialgebraic functions near their points of discontinuity. It turns out to be just as easy to work in the more general setting of o-minimal expansions of \overline{ℝ} and prove stronger statements. In doing so, we establish some results in o-minimality (see 1.1 through 1.8) that seem to be new to the literature even as semialgebraic or subanalytic geometry. We also prove a generalization and some variants of Theorem A, one of which we state now, leaving others for later.

A structure on ℝ is exponentially bounded if for every definable f: ℝ → ℝ there exists m ∈ ℕ such that f is bounded at +∞ by the m-th compositional iterate exp_{m} of exp. It is an easy consequence of quantifier elimination that \overline{ℝ} itself is o-minimal and exponentially bounded.

Theorem B. There is a Cantor set K such that (\overline{ℝ}, K) defines a Borel isomorph of (\overline{ℝ}, ℕ) and, for every exponentially bounded o-minimal expansion ℜ of \overline{ℝ}, every unary set definable in (ℜ, K) either has interior or is Hausdorff null.

1For our purposes, a Cantor set is a subset of ℝ that is nonempty, compact, and has neither interior nor isolated points.

For the following reasons, we regard Theorem B as a natural extension of Theorem A and its original motivating question. (i) By cell decomposition, every o-minimal expansion of the real line (ℝ, <) is interdefinable with a structure on ℝ generated by a collection of open U_{α} ⊆ ℝ^{n(α)}, α ranging over some index set. (ii) For any expansion of (ℝ, <), if every unary definable set either has interior or is Hausdorff null, then every unary definable set either has interior or is nowhere dense. (For every A ⊆ ℝ and open interval I, at most one of I ∩ A and I \ A is Hausdorff null.) (iii) By growth dichotomy [12], Pfaffian closure [17], and Lion et al. [10], if ℜ is an exponentially bounded o-minimal expansion of \overline{ℝ}, then so is (ℜ, exp). In particular, (\overline{ℝ}, exp) is o-minimal and exponentially bounded. (iv) There are now many examples of expansions of \overline{ℝ} that are known to be exponentially bounded and o-minimal.^{2}

It is easy to see that a Cantor set is interdefinable over \overline{ℝ} with the set of midpoints of its complementary intervals (see 2.3), so Theorem B holds with “discrete”, hence also “countable”, in place of “Cantor”, with similar modifications to Theorem A and its corollary.

As of this writing, every expansion of \overline{ℝ} known to be o-minimal is exponentially bounded, thus partly justifying our decision to postpone the statement of our most general version (Theorem C in §2). Another reason is simply to avoid for now having to introduce further notation and technical definitions.

Though one might be tempted to regard both Theorems A and B as teratological, we believe that the techniques of the proofs are interesting in their own right, and potentially useful for other settings.

Here is an outline of the remainder of this paper. We begin in Section 1 with some preliminaries, including some results in pure o-minimality that we believe will be useful in other settings. We prove Theorem B in Section 2, as well as some variants and corollaries. We close in Section 3 with discussion and open issues.

1. Preliminaries

We begin by establishing some global conventions and notation. Throughout, “definable” (in some first-order structure) means “definable with parameters”, while “∅-de-finable” means “definable without parameters”. The variables j, k, m, n range over ℕ, the non-negative integers. Given a set A, its n-fold cartesian power is denoted by A^{n}, with A^{0} := {0}. Whenever convenient, we identify A^{m} × A^{n} with A^{m+n}, in particular, A^{m} × A^{0} ≅ A^{0} × A^{m} ≅ A^{m}. If A belongs to a topological space, we denote its interior by int(A), closure by cl(A), and frontier by fr(A) := cl(A)\A. If A ⊆ ℝ^{n}, then all of these sets (in the usual topology) are ∅-definable in (ℝ, <, A). Given a set B, we identify a function f: A^{0} → B with the constant f(0) ∈ B. Given a function f: A → B and A′ ⊆ A, we let f↾A′ denote the restriction of f to A′. Limits of functions are always taken with respect to the declared domain of the function, and similarly with limits superior and inferior. Metric notions are taken with respect to the sup norm |x| := sup{|x_{1}| , …, |x_{n}|}. In particular, for x ∈ ℝ^{n} and r > 0, put B(x, r) = { y ∈ ℝ^{n} : |x − y| < r }.

Next are some crucial technical results in o-minimality that, as far as we know, are new to the literature even as semialgebraic or subanalytic geometry.

2There are also exponentially bounded expansions of \overline{ℝ} that are not o-minimal. Such structures are closely related to o-minimality in a certain way, but cannot be obtained by expanding \overline{ℝ} by collections of constructible sets (of any arities), and thus are irrelevant for present purposes. See [15] for details.

1.1. Let ℜ be an o-minimal expansion of (ℝ, <), C ⊆ ℝ^{n} be a bounded open cell, and f: C → ℝ be definable, continuous, and bounded. Then there is a definable X ⊆ fr(C) such that dim(fr(C) \ X) < n − 1 and f extends continuously to C ∪ X.

(Recall that dim ∅ = −∞ by convention.)

Note. The resulting extension of f to C ∪ X is necessarily definable, as it is given by y ↦ lim_{x→y} f(x).

Proof. As is often the case in o-minimality, we find it convenient to prove simultaneously a related condition. We proceed by induction on n ≥ 1 to show the following in turn.

(i_{n}) There is a definable Y ⊆ fr(C) such that dim(fr(C) \ Y) < n − 1 and C is locally

connected at every y ∈ Y. (ii_{n}) There is a definable X ⊆ fr(C) such that dim(fr(C) \ X) < n − 1 and f extends

continuously to C ∪ X. (A subset A of a topological space X is locally connected at x ∈ X if for every open neighborhood U of x there is an open neighborhood V ⊆ U of x such that V ∩ A is connected.)

If n = 1, then C is an open interval, so the result is immediate from the monotonicity theorem. Let n ≥ 1 and assume the result for n. Let C ⊆ ℝ^{n+1} be a bounded open cell.

(i_{n+1}). Let D be the projection of C on the first n variables. Then D is a bounded open cell and there exist bounded definable continuous functions g, h: D → ℝ such that g < h and C = { (x, r) : x ∈ D & g(x) < r < h(x) }. Inductively, there exist definable Z ⊆ fr(D) and definable continuous G, H: D ∪ Z → ℝ such that dim(fr(D) \ Z) < n − 1, D is locally connected at every z ∈ Z, g = G↾D, and h = H↾D. Put

Y = graph(g) ∪ graph(h) ∪ { (z, r) : z ∈ Z & G(z) < r < H(z) }.

Then fr(C) \ Y ⊆ fr(graph(g)) ∪ fr(graph(h)) ∪ [(fr(D) \ Z) × ℝ], so dim(fr(C) \ Y) < n. We now show that C is locally connected at every y ∈ Y. As D is open and cells are connected, C is locally connected at every point of graph(g) ∪ graph(h). Let z ∈ Z and r ∈ ℝ be such that G(z) < r < H(z). Let U be an open set containing (z, r). We must find an open box about the point (z, r) that is contained in U and whose intersection with C is connected. By continuity of G and H, there is an open box B × I ⊆ U about (z, r) such that (B × I) ∩ C is disjoint from graph(g) ∪ graph(h). Since D is locally connected at z, we may shrink B so that B ∩ D is connected. Then (B × I) ∩ C is connected, because (B × I) ∩ C = (B ∩ D) × I.

(ii_{n+1}). Let f: C → ℝ be definable, continuous, and bounded. Let Z be the set of all z ∈ fr(C) such that lim_{x→z} f(x) exists. We claim that dim(fr(C) \ Z) < n. Suppose not. Then there is a cell E ⊆ fr(C) such that dim E = n and lim inf_{x→y} f(x) < lim sup_{x→y} f(x) for every y ∈ E. By (i_{n+1}), we may shrink E so that C is locally connected at every y ∈ E. Then fr(graph(f)) ⊇ { (y, r) : y ∈ E & lim inf_{x→y} f(x) < r < lim sup_{x→y} f(x) }, yielding the absurdity

dim fr(graph(f)) ≥ dim E + 1 = n + 1 = dim graph(f) > dim fr(graph(f)).

(See van den Dries [3, pp. 65, 68].) Define g: Z → ℝ by g(z) = lim_{x→z} f(x). Let X be the set of points of continuity of g. By cell decomposition, dim(Z \ X) < dim Z, so dim(fr(C) \ X) < n. Finally, define h: C ∪ X → ℝ by h↾C = f and h↾X = g↾X. Then h is continuous and extends f. □

Remarks. (i) The result and its proof hold for all abstract o-minimal structures (as defined in [3]) provided that the definition of locally connected is relativized to definable connectedness. (ii) If C = { (x, y) ∈ ℝ^{2} : 0 < y < x < 1 } and f(x, y) = y/x, then f is bounded and does not extend continuously to the origin. (iii) If

C = \{ (x, y, z) ∈ ℝ^{3} : |x| < 1 & 0 < y < 1 & −1 < z < \sqrt{|x|}/y \},

then C is bounded and not locally connected at any point of { (0, 0, z) : 0 < z < 1 }.

We need yet more refined results, for which we require some technical definitions and notation.

1.2. Define the corners in ℝ^{n} inductively as follows.^{3} (i) ℝ^{0} is the only corner in ℝ^{0}. (ii) If C ⊆ ℝ^{n} is a corner and f: C → (0, ∞) is continuous, then

{ (x, t) ∈ ℝ^{n+1} : x ∈ C & 0 < t < f(x) }

is a corner in ℝ^{n+1}. We note some easy facts. Every corner in ℝ^{n} is an open cell contained in (0, ∞)^{n}. The projection on the first m coordinates of a corner in ℝ^{m+n} is a corner in ℝ^{m}. For every cell decomposition C of ℝ^{n} that is compatible with (0, ∞)^{n} there is a unique C ∈ C such that C is a corner.

1.3. We now define some special corners. Let Φ be the collection of all homeomorphisms of [0, ∞). Let S_{n} be the collection of all nonempty sets of the form

{ x ∈ (0, ∞)^{n} : ϕ_{n}(x_{n}) < ϕ_{n−1}(x_{n−1}) < ⋯ < ϕ_{1}(x_{1}) < b }

where b ∈ (0, +∞] and ϕ_{1}, …, ϕ_{n} ∈ Φ. Note that this set is an open cell of the structure (ℝ, <, ϕ_{1}, …, ϕ_{n}). We interpret S_{0} as ℝ^{0}. Let S ∈ S_{n}. Note the easy facts: S is a corner; for every m ≤ n, the projection of S on the first m coordinates belongs to S_{m}; and cl(S) \ [(0, ∞)^{n−1} × ℝ] = {0}^{n}. With b = +∞ and ϕ_{i} = t ↦ 2^{i}t for i = 1, …, n, we obtain the set

𝕊_{n} := \{ x ∈ ℝ^{n} : 0 < x_{n} < \frac{x_{n−1}}{2} < ⋯ < \frac{x_{1}}{2^{n−1}} \} .

Note that 𝕊_{n} is definable in (ℝ, <, +).

1.4. Let C ⊆ ℝ^{n} be a corner.

(a)_{n} If (ℝ, <, +, C) is o-minimal, then there exists S ∈ S_{n} definable in (ℝ, <, +, C) such

that cl(S) ∩ (0, ∞)^{n} ⊆ C. (b)_{n} If f: C → ℝ is bounded and (ℝ, <, +, f) is o-minimal, then there exists S ∈

S_{n} definable in (ℝ, <, +, f) such that S ⊆ C and f↾S is continuous and extends continuously to cl(S).

Proof. The case n = 0 is trivial. We now proceed by induction on n ≥ 1. If n = 1, then C is an open interval (0, s) for some s ∈ (0, ∞]. For (a)_{1}, let S be any (0, r) with r ∈ (0, s). Condition (b)_{1} is immediate from the monotonicity theorem and boundedness of f.

Let n ≥ 1, assume the result for n, and let C ⊆ ℝ^{n+1} be a corner. Let π denote projection on the first n coordinates. We use cell decomposition, 1.2, and 1.3 often.

We establish (a)_{n+1}. Suppose that (ℝ, <, +, C) is o-minimal. We produce S ∈ S_{n+1} definable in (ℝ, <, +, C) such that cl(S) ∩ (0, ∞)^{n+1} ⊆ C. It suffices to consider the case that C is bounded. Then πC is a bounded definable corner and there is a continuous definable f: πC → (0, ∞) such that

3More precisely, we define the corners “at 0^{+}”. Corners can be defined relative to any point in [−∞, ∞]^{n} regarded with a fixed appropriate sign condition, but we shall not need any of these variants in this paper.

C = { (x, x_{n+1}) ∈ ℝ^{n+1} : x ∈ πC & 0 < x_{n+1} < f(x) }.

By (a)_{n}, there is a definable S′ ∈ S_{n} such that cl(S′) ∩ (0, ∞)^{n} ⊆ πC. The projection of S′ on the last coordinate is an open interval (0, a), and for every t ∈ (0, a), the set { x ∈ cl(S′) : x_{n} = t } is compact. As f is continuous on cl(S′) ∩ (0, ∞)^{n}, for all t ∈ (0, a) we have 0 < inf{ f(x) : x ∈ cl(S′) & x_{n} = t } < ∞. Define g: (0, a) → ℝ by

g(t) = min(t, inf{ f(x)/2 : x ∈ cl(S′) & x_{n} = t }).

Note that g is definable. As g > 0 and lim_{t→0^{+}} g(t) = 0, there exists by the monotonicity theorem some a′ ∈ (0, a] such that g↾(0, a′) is continuous and strictly increasing. The set { x ∈ S′ : x_{n} < a } is a corner, so inductively, we shrink S′ so that g is continuous and strictly increasing. Put S = { (x, x_{n+1}) : x ∈ S′ & x_{n+1} < g(x_{n}) }. Now,

cl(S) ∩ (0, ∞)^{n+1} ⊆ { (x, x_{n+1}) : x ∈ cl(S′) ∩ (0, ∞)^{n} & 0 < x_{n+1} ≤ f(x)/2 } ⊆ C,

so it suffices to show that S ∈ S_{n+1}. Extend g to ϕ ∈ Φ by setting ϕ(0) = 0 and ϕ(t) = t − a + g(a) for t > a. Write S′ = { x ∈ (0, ∞)^{n} : ϕ_{n}(x_{n}) < ⋯ < ϕ_{1}(x_{1}) < b } as in 1.3. Then ϕ_{n} ∘ ϕ^{−1} ∈ Φ and

S = { (x, x_{n+1}) ∈ (0, ∞)^{n+1} : ϕ_{n} ∘ ϕ^{−1}(x_{n+1}) < ϕ_{n}(x_{n}) < ⋯ < ϕ_{1}(x_{1}) < b } ∈ S_{n+1}.

We establish (b)_{n+1}. Let f: C → ℝ be bounded and (ℝ, <, +, f) be o-minimal. We produce S ∈ S_{n+1} definable in (ℝ, <, +, f) such that S ⊆ C and f↾S is continuous and extends continuously to cl(S). By cell decomposition and (a)_{n+1}, we reduce to the case that C ∈ S_{n+1} and f is continuous and extends continuously to cl(C) ∩ (0, ∞)^{n+1}. By 1.1 and (a)_{n+1}, we may shrink πC (hence also C) so that there is a definable continuous g: C ∪ (πC × {0}) → ℝ with f = g↾C. By (b)_{n}, we reduce to the case that g↾(πC × {0}) extends continuously and definably to cl(πC) × {0}. By replacing f with f − g(πx, 0), we reduce to the case that g↾(πC × {0}) = 0. Thus, we now need only find S ∈ S_{n+1} definable in (ℝ, <, +, f) such that S ⊆ C and lim_{x→0}(f↾S)(x) = 0, which is immediate from (a)_{n+1} and that the set { x ∈ C : |f(x)| ≤ x_{n} } contains a corner: For each fixed x ∈ C, we have x_{n} > 0 and lim_{t→0^{+}} f(πx, t) = 0, so there exists ϵ > 0 such that, for all t ∈ (0, ϵ), we have (πx, t) ∈ C and |f(πx, t)| ≤ x_{n}. □

1.5. Let T_{n} be the group of symmetries, regarded as linear transformations ℝ^{n} → ℝ^{n}, of the polyhedron inscribed in the unit ball in ℝ^{n} whose vertices are the intersections of the unit sphere in ℝ^{n} with the set { tu : t > 0 & u ∈ {−1, 0, 1}^{n} }. For example, T_{2} is the symmetry group of the octagon. With 𝕊_{n} as in 1.3, we have ℝ^{n} = ⋃_{T ∈T_{n}} T (cl(𝕊_{n})) = ⋃_{T ∈T_{n}} cl(T (𝕊_{n})).

As an immediate consequence of cell decomposition, 1.2, and 1.4, we obtain a key technical lemma:

1.6. Let A ⊆ ℝ^{n} and f: A → ℝ be such that (ℝ, <, +, f) is o-minimal. Let f be bounded near y ∈ cl(A). Then there exists S ∈ S_{n} definable in (ℝ, <, +, f) such that for every T ∈ T_{n} and m ≤ n, the restriction of f to A ∩ (y + T (π_{m}S × {0}^{n−m})) is continuous and extends continuously to the closure, where π_{m} denotes projection on the first m coordinates.

Remark. The result is easily modified to hold for o-minimal expansions of arbitrary ordered groups. As one might imagine, the result can be generalized significantly if ℜ also expands an ordered field—recall the triangulation theorem—but it takes a bit of effort to make this precise. As we shall not need any of these of these generalizations in this paper, we leave details to the interested reader.

1.7. We now define some special elements of S_{n}. First, define ψ: [0, ∞) → ℝ by

ψ(t) = \begin{cases}0, & t = 0 \\ e^{−1/t}, & 0 < t < 1 \\ t − 1 + e^{−1}, & t ≥ 1. \end{cases}

Note that ψ ∈ Φ. For l ∈ ℤ, let ψ_{l} be the l-th compositional iterate of ψ. Every ψ_{l} is definable in (\overline{ℝ}, exp). For l ≥ 0, ψ_{l}(t) = 1/ exp_{l}(1/t) for all sufficiently small t > 0, and ψ_{−l}(t) = 1/ log_{l}(1/t) for all sufficiently small t > 0, where log_{l} denotes the ultimately defined l-th compositional iterate of log. If ℜ is an exponentially bounded expansion of \overline{ℝ}, then for each definable f: (0, b) → (0, ∞) such that lim_{t→0^{+}} f(t) = 0 there is a least l ∈ ℤ such that f(t) ≥ ψ_{l}(t) as t → 0^{+}; if ℜ is also o-minimal, then f(t) < ψ_{l−1}(t) as t → 0^{+}. Define sets S_{n,l} ∈ S_{n} by S_{n,l} = { x ∈ ℝ^{n} : 0 < x_{n} < ψ_{l}(x_{n−1}) < ⋯ < ψ_{(n−1)l}(x_{1}) }. Every S_{n,l} is an open cell of (\overline{ℝ}, exp). An easy induction shows that if S ∈ S_{n} and (\overline{ℝ}, S) is o-minimal and exponentially bounded, then there exist δ, l > 0 such that B(0, δ)∩S_{n,l} ⊆ S. Hence, by 1.6,

1.8. Let A ⊆ ℝ^{n} and f: A → ℝ be such that (\overline{ℝ}, f) is o-minimal and exponentially bounded. Let f be bounded near y ∈ cl(A). Then there exist δ, j > 0 such that for every T ∈ T_{n} and m ≤ n, the restriction of f to B(y, δ) ∩ A ∩ (y + T (S_{m,j} × {0}^{n−m})) is continuous and extends continuously to the closure.

Remark. The above holds with “polynomially” instead of “exponentially” by using the sets

{ x ∈ ℝ^{m} : 0 < x_{n} < x_{n−1}^{l} < ⋯ < x_{1}^{l^{m−1}} }

instead of the S_{n,l}. Our results so far have not required working over ℝ in that they are easily generalized to abstract o-minimal structures. This now begins to change.

Given A ⊆ ℝ^{n} and r > 0, let N(A, r) be the infimum of all k such that A is covered by k-many closed cubes of side length r. The set A is Minkowski null^{4} if lim_{r→0^{+}} r^{ϵ}N(A, r) = 0 for all ϵ > 0. While Minkowski nullity is not generally preserved under countable unions or C^{1} images, it is much better behaved than Hausdorff nullity in some other ways. We extend this notion relative to expansions ℜ of \overline{ℝ} by defining A ⊆ ℝ^{n} to be ℜ-null if lim_{r→0^{+}} f(r)N(A, r) = 0 for every definable f: ℝ^{>0} → ℝ such that lim_{r→0^{+}} f(r) = 0. As \overline{ℝ} defines all rational power functions, ℜ-null implies Minkowski null. If ℜ is polynomially bounded, then ℜ-null is the same as Minkowski null. We have some other easy basic facts that will be used often.

1.9. (1) ℜ-nullity is preserved under taking subsets, closure, finite unions, finite carte-

sian products, and images under Lipshitz maps.

(2) ℜ-null sets are nowhere dense and Hausdorff null.

(3) Countable unions of ℜ-null sets are Baire meager and Hausdorff null.

4Also known under several other names and equivalent formulations. See [11, Ch. 5] for more information.

(4) If every bounded unary definable set either has interior or is ℜ-null, then every unary definable set either has interior or is Hausdorff null.

(5) If ℜ is o-minimal and exponentially bounded, then A ⊆ ℝ^{n} is ℜ-null if and only if lim_{r→0^{+}} r exp_{m}(N(A, r)) = 0 for all m.

Proof. We give a sketch for (5) but leave the rest as exercises.

Suppose that A is ℜ-null. Then lim_{r→0^{+}} N(A, r)/ log_{m}(ϵ/r) = 0 for every m ∈ ℕ and ϵ > 0. Hence, N(A, r) < log_{m}(ϵ/r) for all sufficiently small r > 0. Then r exp_{m}(N(A, r)) < ϵ for all sufficiently small r > 0.

Conversely, suppose that lim_{r→0^{+}} r exp_{m}(N(A, r)) = 0 for all m. We need only show that lim_{r→0^{+}} N(A, r)/ log_{m}(1/r) = 0 for all m. By assumption, lim_{r→0^{+}} r exp_{m+1}(N(A, r)) = 0, so N(A, r) < log_{m+1}(1/r) for all sufficiently small r > 0. Now note that

lim_{r→0^{+}} log_{m+1}(1/r)/ log_{m}(1/r) = 0.  □

1.10. Let ℜ be an o-minimal expansion of \overline{ℝ}. If A ⊆ ℝ^{m} is ℜ-null, B ⊆ ℝ^{m} is compact, and f: B → ℝ^{n} is continuous and definable, then f(A ∩ B) is ℜ-null.

Proof. We may take A ⊆ B. Let g: ℝ^{>0} → ℝ be definable such that lim_{r→0^{+}} g(r) = 0. We must show that lim_{r→0^{+}} g(r)N(f(A), r) = 0. It suffices to consider the case that g ≥ 0. By generalized Hölder continuity [4, C.15], there is a definable ϕ ∈ Φ (recall 1.3) such that |f(x) − f(y)| ≤ ϕ(|x − y|) for all x, y ∈ B; then N(f(A), ϕ(r)) ≤ N(A, r) for all r > 0. As g ∘ ϕ is definable and A is ℜ-null, we have lim_{r→0^{+}} g(ϕ(r))N(A, r) = 0, hence also

0 = lim_{r→0^{+}} g(ϕ(r))N(f(A), ϕ(r)) = lim_{r→0^{+}} g(r)N(f(A), r).  □

We next recall a result from [5] and some minor variants. Given a structure ℜ on ℝ and Y ⊆ ℝ, let (ℜ, Y)^{#} denote the expansion (ℜ, (X)) of ℜ, where X ranges over all subsets of all cartesian powers of Y.

1.11. Let A ⊆ ℝ and ℜ be an o-minimal expansion of (ℝ, <, +, 1). Let B ⊆ ℝ have no interior and be definable in (ℜ, A)^{#}. Then there exists f: ℝ^{n} → ℝ definable in ℜ such that B ⊆ cl(f(A^{n})). If B is bounded, then f can be taken to be bounded by replacing it with max(N, |f|) for some N such that B ⊆ [−N, N]. This all holds with “∅-definable” in place of “definable”.

Thus, as ℜ-nullity is preserved under taking closure, we have

1.12. Let ℜ be an o-minimal expansion of \overline{ℝ} and A ⊆ ℝ be such that f(A^{n}) is ℜ-null for all bounded definable f: ℝ^{n} → ℝ. Then every bounded unary set definable in (ℜ, A)^{#} either has interior or is ℜ-null.

We leave the proof of the following amusing result as an exercise.

1.13. For every ℚ-linearly independent A ⊆ ℝ, the function x ↦ ∑_{i=1}^{n} 2^{i−1}x_{i}: ℝ^{n} → ℝ is injective on A^{n}.

Next is a combination of basic definability and descriptive set theory.

1.14. Let A ⊆ ℝ be uncountable and Σ_{1}^{1}. Then there is a Cantor set K such that: (1) (ℝ, <, +, 1, K) ∅-defines a Borel isomorph of (\overline{ℝ}, ℕ).

(2) For every bounded f: ℝ^{n} → ℝ there is a finite set H of bounded functions ∅-defin-able in (ℝ, <, +, 1, f) such that f(K^{n}) ⊆ ⋃_{h∈H} h(A^{n(h)}).

Proof. There exists l ∈ ℤ such that A ∩ [l, l + 1] is uncountable. By translation, we take A ⊆ [1, 2]. Every uncountable Σ_{1}^{1} set contains a Cantor set, so we take A to be Cantor. Every Cantor set contains a ℚ-linearly independent Cantor set (use [9, 19.1]), so we also take A to be ℚ-linearly independent. Let D be the result of removing from A its maximum and minimum, and all left endpoints of the complementary intervals of A. By a classical construction—for example, Gelbaum and Olmstead [7, I.8.14]—there is a strictly increasing bijection g: ℝ → D whose compositional inverse is continuous. Hence, g is a Borel isomorphism and a closed map. Put

X = { (g(x), g(y), g(x + y), g(xy), g(d_{ℕ}(x))) : x, y ∈ ℝ }

where d_{ℕ}: ℝ → ℝ denotes the distance function to ℕ. Define T: ℝ^{5} → ℝ by T (x) = ∑_{i=1}^{5} 2^{i−1}x_{i}. Put K = A ∪ T (cl(X)). As X has no isolated points, the same is true of cl(X), hence also of T (cl(X)). Since cl(X) ⊆ A^{5} and T ↾A^{5} is injective (1.13), T (cl(X)) is compact and has no interior. Thus, T (cl(X)) is a Cantor set, hence so is K.

For (1), observe that (ℝ, X) ∅-defines the image under g of (\overline{ℝ}, d_{ℕ}), hence also that of (\overline{ℝ}, ℕ), so it suffices to show that (ℝ, < +, 1, K) ∅-defines X. We have cl(X) ⊆ A^{5} ⊆ [1, 2]^{5} and T ([1, 2]^{5}) ⊆ (2, ∞), so K is the disjoint union of A and T (cl(X)). Thus, A = K ∩ [1, 2] and (ℝ, <, +, 1, K) ∅-defines A, hence also D. As g is a closed map and T ↾A^{5} is injective, we also have X = D^{5} ∩ T^{−1}(K \ A).

Finally, (2) is straightforward, but tedious to write up in detail. We illustrate the point via the case n = 2 and leave the rest to the reader. Since K ⊆ A ∪ T (A^{5}), we have

K^{2} ⊆ A^{2} ∪ A × T (A^{5}) ∪ T (A^{5}) × A ∪ (T (A^{5}))^{2}.

Let f: ℝ^{2} → ℝ be bounded. Put H = {h_{1}, …, h_{4}}, where

h_{1} = f
h_{2} = f(x_{1}, T (x_{2}, …, x_{6}))
h_{3} = f(T (x_{1}, …, x_{5}), x_{6})
h_{4} = f(T (x_{1}, …, x_{5}), T (x_{6}, …, x_{10})).

Then f(K^{2}) ⊆ h_{1}(A^{2}) ∪ h_{2}(A^{6}) ∪ h_{3}(A^{6}) ∪ h_{4}(A^{10}). □

Remark. By using distance functions, the set X is easily modified to encode over (ℝ, <, +, 1) any expansion of (ℝ, <, +, 1) by finitely many closed sets.

2. Main results

In this section, we prove Theorem B, as well as some variants and corollaries.

Proof of Theorem B. We must produce a Cantor set K such that:

• (\overline{ℝ}, K) defines a Borel isomorph of (\overline{ℝ}, ℕ).

• For every exponentially bounded o-minimal expansion ℜ of \overline{ℝ}, every unary set definable in (ℜ, K) either has interior or is Hausdorff null.

By diagonalization, we fix a sequence (r_{k}) of positive real numbers such that r_{0} = 1, 2r_{k+1} < r_{k} for all k, and lim_{k→+∞} r_{k+1}/ψ_{j}(r_{k}) = 0 for all j (ψ_{j} as in 1.7). Define sets E_{k} inductively by E_{0} = [0, 1] and E_{k+1} = E_{k} \ ⋃_{c}(c + r_{k+1}, c + r_{k} − r_{k+1}), where c ranges over the left endpoints of the connected components of E_{k}. Then ⋂_{k} E_{k} is a Cantor set. Let E be a ℚ-linearly independent Cantor set contained in ⋂_{k} E_{k} (recall the proof of 1.14). Let K be a Cantor set constructed as in 1.14 with A = E; then (\overline{ℝ}, K) defines a Borel isomorph of (\overline{ℝ}, ℕ).

Let ℜ be an exponentially bounded o-minimal expansion of \overline{ℝ}. We show that every unary set definable in (ℜ, K) either has interior or is Hausdorff null. Indeed, with an eye toward applications and further generalization, we show that every bounded unary set definable in (ℜ, K)^{#} either has interior or is ℜ-null (recall 1.9.4). We use repeatedly that ℜ-nullity is preserved under finite unions. By 1.12 and 1.14, it suffices to let f: ℝ^{n} → ℝ be bounded and definable in ℜ, and show that f(E^{n}) is ℜ-null. By compactness of E^{n}, it is enough to show that if x ∈ E^{n} then f(E^{n} ∩ B(x, δ)) is ℜ-null for some δ > 0. By 1.8, there exist δ, j > 0 such that for every T ∈ T_{n} and m ≤ n, the restriction of f to (x + T (S_{m,j} × {0}^{n−m})) ∩ B(x, δ) is continuous and extends continuously to the closure. Thus, it suffices now by 1.10 to show that:

(i) Every E^{n} is ℜ-null.

(ii) For every j > 0 there exists δ > 0 such that for all n,

E^{n} − E^{n} ∩ (−δ, δ)^{n} ⊆ ⋃_{m≤n} ⋃_{T ∈T_{n}} T (S_{m,j} × {0}^{n−m}),

where E^{n} − E^{n} denotes the difference set of E^{n}.

Proof of (i). As ℜ-nullity is preserved under finite cartesian products, it suffices by 1.9.5 to fix m and show that lim_{r→0^{+}} r exp_{m}(N(E, r)) = 0. Let 0 < r < 1; then there exists k such that r_{k+1} ≤ r < r_{k}. For every j, the set E_{j} consists of 2^{j} disjoint closed intervals each of length r_{j}, so r exp_{m}(N(E, r)) ≤ r exp_{m}(2^{k+1}) < r_{k} exp_{m+1}(k + 1). If r is sufficiently small, then r_{k−1} ≤ 1/(k +1) and exp_{m+1}(k +1) = 1/ψ_{m+1}(1/(k +1)), so r exp_{m}(N(E, r)) ≤ r_{k}/ψ_{m+1}(r_{k−1}). By construction, lim_{k→+∞} r_{k}/ψ_{m+1}(r_{k−1}) = 0.

Proof of (ii). As j > 0, we have t−ψ_{j}(t) > 0 for all sufficiently small t > 0. By exponential bounds and properties of (r_{k}), we have

lim_{k→+∞} \frac{r_{k+1}}{r_{k} − ψ_{j}(r_{k})} = 0 = lim_{k→+∞} \frac{r_{k+1}}{ψ_{j}(ψ_{j}(r_{k}))}.

Thus, there exists N ∈ ℕ such that for all k > N we have r_{k+1} < ψ_{j}(ψ_{j}(r_{k})) and ψ_{j}(r_{k}) < r_{k} − r_{k+1}, hence also r_{k+1} < ψ_{j}(r_{k} − r_{k+1}). Put δ = r_{N}. We now proceed by induction on n. The case n ≤ 1 is trivial. Let n > 1 and assume the result for all lower values of n. The argument is routine, but a bit tedious to write up in detail; we give only an outline. By permuting coordinates, it is enough to show that

E^{n} − E^{n} ∩ (0, δ)^{n} ⊆ ⋃_{m≤n} ⋃_{T ∈T_{n}} T (S_{m,j} × {0}^{n−m}).

By ℚ-linear independence and symmetry, it is enough to show that

E^{n} − E^{n} ∩ (0, δ)^{n} ∩ 𝕊_{n} ⊆ S_{n,j}

(recall 1.3 and 1.5). Let (x, x_{n−1}, x_{n}), (y, y_{n−1}, y_{n}) ∈ E^{n−2}×E×E be such that (x, x_{n−1}, x_{n})− (y, y_{n−1}, y_{n}) ∈ (0, δ)^{n} ∩ 𝕊_{n}. Inductively, (x, x_{n−1}) − (y, y_{n−1}) ∈ S_{n−1,j}, so it suffices to show that x_{n} −y_{n} < ψ_{j}(x_{n−1} −y_{n−1}). Let k be such that r_{k+1} < x_{n−1} −y_{n−1} ≤ r_{k}. It suffices now by choice of δ and monotonicity of ψ_{j} to show that x_{n}−y_{n} ≤ r_{k+1}. By construction of E, we have E^{2}∩[0, r_{k}]^{2}∩𝕊_{2} ⊆ [0, r_{k+1}]^{2}∪[r_{k}−r_{k+1}, r_{k}]×[0, r_{k+1}]. Since x_{n−1}−y_{n−1} ∈ [r_{k}−r_{k+1}, r_{k}], we have x_{n} − y_{n} ∈ [0, r_{k+1}]. □

Having established Theorem B, we now proceed to some variants and corollaries.

Following [6], we say that a sequence (a_{k}) of positive real numbers is fast for an expansion ℜ of \overline{ℝ}, or ℜ-fast, if lim_{k→+∞} f(a_{k})/a_{k+1} = 0 for every f: ℝ → ℝ definable in ℜ. For (r_{k}) as in the proof of Theorem B, the sequence (1/r_{k}) is fast for every exponentially bounded expansion of \overline{ℝ}. An examination of the proof of Theorem B yields the following generalization.

Theorem C. Let (a_{k}) be a sequence of positive real numbers. Then there is a Cantor set K such that (ℝ, <, +, 1, K) ∅-defines a Borel isomorph of (\overline{ℝ}, ℕ), and for every o-minimal expansion ℜ of \overline{ℝ}, if (a_{k}) is ℜ-fast, then every bounded unary set definable in (ℜ, K)^{#} either has interior or is ℜ-null.

Under fairly reasonable assumptions, ℜ-fast sequences exist.

2.1. Let ℜ be an expansion of \overline{ℝ}.

(1) If there is a countable collection F of functions ℝ → ℝ such that every unary function definable in ℜ is bounded at +∞ by a member of F, then there exist ℜ-fast sequences.

(2) If ℜ is o-minimal and the language of ℜ is countable, then there exist ℜ-fast sequences.

Proof. Diagonalization yields (1). For (2), suppose that ℜ is o-minimal. By the proof of [4, C.4], every unary function definable in ℜ is bounded at +∞ by a unary function ∅-definable in ℜ. Since the language is countable, there are only countably many unary functions ∅-definable in ℜ. Apply (1). □

When combined with Theorem C,

2.2. Let (ℜ_{k})_{k∈ℕ} be a sequence of o-minimal expansions of \overline{ℝ}, each in a countable language. Then there is a Cantor set K such that (\overline{ℝ}, K) defines a Borel isomorph of (\overline{ℝ}, ℕ) and every bounded unary set definable in any (ℜ_{k}, K)^{#} either has interior or is Minkowski null.

2.3. Theorem C holds with “discrete”, hence also “countable”, in place of “Cantor”.

Proof. Let E be any Cantor set and M be the set of midpoints of the (bounded) complementary intervals of E. Note that

M = { r ∈ ℝ : ∃ϵ > 0, E ∩ [r − ϵ, r + ϵ] = {r − ϵ, r + ϵ} }

and E = fr(M). Hence, (\overline{ℝ}, E) and (\overline{ℝ}, M) are ∅-interdefinable. □

We now answer a question raised in [13, §3.1]. A set A ⊆ ℝ^{n} has a locally closed point if it has nonempty interior in its closure. It is easy to see that if every nonempty unary set definable in ℜ has a locally closed point, then every definable unary set either has interior or is nowhere dense. We show that the converse fails.

2.4. There exist ∅ ≠ A ⊆ ℝ having no locally closed points such that every unary set definable in (\overline{ℝ}, A) either has interior or is nowhere dense.

Proof. With K as in Theorem B, let A be the set of left endpoints of the complementary intervals of K. Note that A ⊆ K and cl(A) = cl(K \ A) = K. □

3. Discussion and open issues

It is easy to construct Cantor sets E that do define ℕ over \overline{ℝ}: Just encode ℕ by the set of lengths of the complementary intervals. For example, remove successively from [0, 1] the middle intervals of length 1/(n!) for n ≥ 2. Then (\overline{ℝ}, E) defines the set A := \{ n! : n ∈ ℕ \}, hence also the successor function σ: A → A, hence also ℕ = {0} ∪ { σ(a)/a : a ∈ A }.

Though Theorem A answers one question, others arise immediately. Let us consider a few of them.

Are there constructible E ⊆ ℝ and N ∈ ℕ such that (\overline{ℝ}, E) defines a nonconstructible set and every set definable in (\overline{ℝ}, E) is Σ_{N}^{1}? We believe so. The proof of 2.4 shows that if E is a Cantor set, then (\overline{ℝ}, E) defines a unary set that is not F_{σ}, hence not constructible. We think there are Cantor sets E such that every set definable in (\overline{ℝ}, E) is a boolean combination of F_{σ} sets. Along these lines, the Cantor set K in 1.14 was designed to encode (\overline{ℝ}, ℕ). But we could omit deliberately encoding ℕ, that is, in the proof of 1.14, replace X with its projection on the first four coordinates, then define K as in the proof of Theorem B, and so on. What can be said about the definable sets of (\overline{ℝ}, K)? (Of course, by the rest of the proof of Theorem B, every unary definable set either has interior or is Hausdorff null.) Or, in the definition of X, replace d_{ℕ} with exp. Then (\overline{ℝ}, K) Borel-interprets (\overline{ℝ}, exp). Is exp definable? And so on. The technology from [5] appears not to tell us much more about the sets definable in (\overline{ℝ}, E) than those in (\overline{ℝ}, E)^{#}, so it seems that new ideas are needed.

Are there constructible E ⊆ ℝ and N ∈ ℕ such that (\overline{ℝ}, E) defines a non-Borel set (or even just of infinite Borel rank), yet every set definable in (\overline{ℝ}, E) is Σ_{N}^{1}? (If so, then N ≥ 2 by Souslin’s Theorem [9, 14.11].) Here, we have no ideas as yet.

Evidently, one can generalize these questions. For example, allow E to be F_{σ}, or even just of finite Borel rank. We can go in the other direction as well. In Theorem A, can we take E to be closed and countable? Closed and discrete? (cf. 2.3.) Regarding the last, there are some known restrictions. If E ⊆ (0, ∞) is infinite, closed and discrete, then it is the range of a strictly monotone and unbounded-above sequence (a_{k}) of positive real numbers. If a_{k+1}/a_{k} → 1, then (\overline{ℝ}, E) defines ℕ by Hieronymi [8]. On the other hand, if (log a_{k+1})/(log a_{k}) → ∞, that is, if (a_{k}) is \overline{ℝ}-fast, then every set definable in (\overline{ℝ}, E) is constructible [6]. See [14] for some related results.

(There are non-Borel E ⊆ ℝ such that every set definable in (\overline{ℝ}, E) is Σ_{2}^{1}, but this is far off the point of this paper, so we give only a hint: By van den Dries [2, Theorem 1], if E is Σ_{N}^{1} and a real-closed subfield of ℝ, then every set definable in (\overline{ℝ}, E) is a boolean combination of Σ_{N}^{1} sets.)

We close with a few words on history and attribution. Recall that our original goal was to produce a constructible E ⊆ ℝ such that (\overline{ℝ}, E) defines a non-constructible set but does not define ℕ. Every expansion of \overline{ℝ} that defines ℕ also ∅-defines it, hence also ℚ, because ℕ is the unique subset of [0, ∞) that is closed under x ↦ x + 1 and whose intersection with [0, 1] is equal to {0, 1}. Hence, the following result due to Friedman and Miller suffices.

Theorem A_{0}. There is a closed E ⊆ ℝ such that (\overline{ℝ}, E) defines a Borel isomorph of (\overline{ℝ}, ℕ) and every unary set ∅-definable in (\overline{ℝ}, E) either has interior or is nowhere dense.

For this, it suffices by 1.11 and 1.14 to find a Cantor set E such that f(E^{n}) is nowhere dense for every f: ℝ^{n} → ℝ that is ∅-definable in \overline{ℝ}. The naive approach described in the introduction appears to stall for functions ∅-definable in \overline{ℝ} just as it does for parametrically definable functions. But in any o-minimal structure in a countable language, there are only countably many ∅-definable functions, and they are all Borel (by cell decomposition). This prompted Friedman to produce the following result of independent interest, thus establishing an appropriately modified version of 2.2, hence also Theorem A_{0}.

3.1. For every sequence (f_{k}: ℝ^{n(k)} → ℝ)_{k∈ℕ} of Borel functions there is a Cantor set E such that every image f_{k}(E^{n(k)}) is nowhere dense.

We shall not prove this here as we no longer need it; the interested reader may wish to attempt verification by amalgamating the proofs of 19.1 and 19.8 from [9]. On the other hand, attempts by Friedman and Miller to derive Theorem A from Theorem A_{0} were unsuccessful, as were attempts to conclude from 3.1 the existence of a Cantor set E such that f(E^{n}) is nowhere dense for every semialgebraic f: ℝ^{n} → ℝ. Of course, 3.1 is a very blunt hammer in this setting, as it uses nothing about o-minimality (yet relies heavily on countability). Convinced that a more singularity-theoretic approach was in order, Miller approached Kurdyka and Speissegger, who subsequently solved the semialgebraic case, which Miller then refined to its current form. The crucial idea of using Minkowksi rather than Hausdorff nullity is due to Kurdyka, who credits Yomdin and Comte [18] for inspiration. Result 1.1 is due to Speissegger, who had known it for several years but had not made prior use of it.

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

E-mail address: friedman@math.ohio-state.edu

Laboratoire de Mathématiques (LAMA), Université de Savoie, UMR 5127 CNRS, 73376 Le Bourget-du-Lac cedex France

E-mail address: Krzysztof.Kurdyka@univ-savoie.fr

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

E-mail address: miller@math.ohio-state.edu

Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

E-mail address: speisseg@math.mcmaster.ca