AN UPGRADE FOR “AVOIDING THE PROJECTIVE HIERARCHY IN

EXPANSIONS OF THE REAL FIELD BY SEQUENCES” (POST HIERONYMI)

CHRIS MILLER

Version: January 7, 2016

The reader is assumed to have [M2] at hand. I give here: (i) stronger versions of some of the results due to a remarkable result of Philipp Hieronymi; (ii) a refinement of 3.1 and an application thereof, and (iii) an alternate formulation of Theorem A and an easier proof (that given in [M2] being based on lemmas needed to prove the rather more general 3.2).

But first, I correct a minor error: In the proof of 1.7, “well ordered” should be “anti well ordered”.

Hieronymi’s Theorem and consequences

Theorem (Hieronymi [H2]). If E ⊆ ℝ is discrete, and f: E^{n} → ℝ is somewhere dense, then (\overline{ℝ}, f) defines ℕ.

As a fairly easy consequence [H1]: If α, β > 0 are such that log α and log β are ℚ-linearly independent, then (\overline{ℝ}, α^{ℤ}, β^{ℤ}) = PH. Hence also: If α > 1 and r ∈ ℝ\ℚ, then (\overline{ℝ}, x^{r}, α^{ℤ}) = PH. Consequently, in the statement of [M2, 1.5], replace “every definable subset of ℝ either has interior or is nowhere dense” with “ℜ does not define ℕ”. As pointed out in [M2, 1.6], the lack of having this knowledge on hand at the time resulted in a number of awkward formulations of results. I shall clean these up below.

Remarks. (i) As yet another consequence of Hieronymi’s theorm, we have a strengthening of the first part of AEG: An expansion of \overline{ℝ} defines ℕ iff it defines the range of a strictly monotone sequence (a_{k})_{k∈ℕ} of nonzero real numbers such that lim_{k→+∞}(a_{k+1}/a_{k}) = 1. This could probably be used to shorten some of proofs in [M2], but as far as I know, it doesn’t extend any results beyond those that are implied by the two previously-mentioned consequences. (ii) Subsequent joint work with Antongiulio Fornasiero and Hieronymi [FHM] and Hieronymi [HM] might result in yet further upgrades, but I have not worked this out.

Direct changes.

The conclusion of 1.11 becomes:

Then:

(a) There exist c > 0 and F ∈ ℝ((x^{ℚ}))_{ω} such that supp(F) ⊆ (−∞, 0], and either supp(F) is infinite and f ∼ c log x + F, or supp(F) is finite and ultimately f = c log x + F.

(b) (\overline{ℝ}, ⟨f^{−1}⟩) defines (e^{1/c})^{ℤ}.

(c) (\overline{ℝ}, f′, ⟨f^{−1}⟩) has field of exponents ℚ.

Some of the text in 1.13 needs obvious updating.

The conclusion of 2.2 becomes:

Then:

(a) There exist c > 0 and F ∈ ℝ((x^{ℚ}))_{ω} such that supp(F) ⊆ (−∞, 0], and either supp(F) is infinite and f ∼ c log x + F, or supp(F) is finite and ultimately f = c log x + F.

(b) (\overline{ℝ}, sin f) defines (e^{π/c})^{ℤ}.

(c) (\overline{ℝ}, sin f) has field of exponents ℚ.

3.4 is omitted, and the conclusion of 3.2 becomes:

Then there exist 0 < α ≠ 1 and F ∈ ℝ((x^{ℚ}))_{ω} such that either supp(F) is infinite and f ∼ F (α^{x}), or supp(F) is finite and exactly one of the following holds:

(1) f = F (α^{x});

(2) f − F (α^{x}) ∉ ℝ and ∥f − F (α^{x})∥ > α^{x^{n}} for every n ∈ ℕ;

(3) f − F (α^{x}) ∉ ℝ and |∥f − F (α^{x})∥ α^{−P} − 1| ≤ cα^{−rx} for some c, r > 0 and P ∈ ℝ[x] of degree at least 2.

A refinement of 3.1 and an application to d-minimality

0.1 (a refinement of 1.4). Let P ∈ ℝ[x] and f be infinitely increasing such that f ∼ e^{P}. Then (\overline{ℝ}, ⟨f⟩) defines e^{βℤ}, where β is the leading coefficient of P.

Proof. Check that a = β(d − 1)! in the proof of 1.4. □

0.2. Let ℜ be an o-minimal expansion of \overline{ℝ}, S_{1}, …, S_{N} ⊆ ℝ be countable sets, and h: ℝ^{N} → ℝ be given. If every subset of ℝ definable in (ℜ, h, S_{1}, …, S_{N}) either has interior or is nowhere dense, or if (ℜ, h, S_{1}, …, S_{N}) is d-minimal, then the same is true of the expansion of ℜ by all subsets of h(S_{1} × ⋯ × S_{N}).

Proof. Immediate from [FM, Theorem B and Claim on pg. 62]. □

0.3 (a refinement of 3.1). If f ∈ H is infinitely increasing, bounded above by some e^{x^{N}}, and (\overline{ℝ}, ⟨f⟩) ≠ PH, then there exist β, c, r > 0 and a monic P ∈ x.ℚ[x] + ℝ such that |fe^{−βP} − 1| ≤ ce^{−rx}.

(A refinement of Theorem B also follows easily; I leave details to the reader.)

Proof. By 3.1, there exists P ∈ ℝ[x] such that the remaining conditions hold, so we need show only that P − P (0) ∈ ℚ[x]. Write P = ∑_{j=0}^{d} a_{j}x^{j}, a_{d} = 1. Put M = min{ m : a_{m}, …, a_{d} ∈ ℚ }; we must show that M = 1. Put Q = ∑_{j=M}^{d} a_{j}x^{j}. Note that

⟨f⟩, ⟨fe^{−βQ}⟩ ⊆ ⟨f⟩ · ∏_{j=M}^{d}⟨e^{βx}⟩^{−a_{j}}.

By 0.1, (\overline{ℝ}, ⟨f⟩) defines ⟨e^{βx}⟩. By 0.2, every subset of ℝ definable in (\overline{ℝ}, ⟨f⟩, ⟨fe^{−βQ}⟩) either has interior or is nowhere dense. Since fe^{−βQ} ∼ e^{β(P −Q)}, we have M = 1 by 0.1 and 1.5. □

0.4 (an application of 0.3). Let α > 1, P ∈ ℝ[x] \ ℝ and ℜ be an o-minimal expansion of \overline{ℝ}. Then (ℜ, ⟨α^{P} ⟩) is d-minimal iff ℜ has field of exponents ℚ and there exists β ∈ ℝ such that P − P (0) ∈ β.ℚ[x].

Proof. It suffices to consider the case α = e, P is monic, and P (0) = 0.

The forward implication is immediate from the definition of d-minimality, 0.1, 1.5 and 0.3. Assume that ℜ has field of exponents ℚ and P ∈ β.ℚ[x] for some β ∈ ℝ. Write

P = β ∑_{j=1}^{d} q_{j}x^{j}, q_{d} = 1. By 0.1, (ℜ, ⟨e^{P} ⟩) defines ⟨e^{βx}⟩. Note that ⟨e^{P} ⟩ ⊆ ∏_{j=1}^{d}⟨e^{βx}⟩^{q_{j}}. By [M1, §3.4], (ℜ, ⟨e^{βx}⟩) is d-minimal. Apply 0.2. □

An alternate version of Theorem A

Here, i := \sqrt{−1}, and for r > 0, x^{ir} denotes the restriction to the positive real line of the complex power function z^{ir}, defined with respect to an appropriate branch of log z.

Recall 2.1.

Theorem A′. Let f: ℝ → ℝ be bounded below as x → +∞ by a compositional iterate of log x. If (\overline{ℝ}, f) is o-minimal and (\overline{ℝ}, e^{if}) does not define ℕ, then there exist r > 0 and c ∈ ℂ\\{0\} such that e^{if} ∼ cx^{ir}. Moreover, (\overline{ℝ}, e^{if}) defines x^{ir}, hence also the group (e^{π/r})^{ℤ}, so (\overline{ℝ}, e^{if}) has field of exponents ℚ.

Proof. Note that (\overline{ℝ}, e^{if}) defines f′ (= (e^{if})′/ie^{if}) and (\overline{ℝ}, f′) is o-minimal. We show that (\overline{ℝ}, f′) defines no h: ℝ → ℝ such that f′ ∼ h′. Suppose otherwise. Then h is infinitely increasing (by L’Hôpital), (\overline{ℝ}, e^{if}) defines h^{−1}, and f ∘ h^{−1} = x + g with g′ → 0. For each t ∈ ℝ, we thus have e^{it} = lim_{x→+∞} e^{if(h^{−1}(t+x))}/e^{if(h^{−1}(x))}, so (\overline{ℝ}, e^{if}) defines e^{ix}, hence also ℕ, a contradiction. As in the proof of 1.11, (\overline{ℝ}, f′) is polynomially bounded and f′ has an asymptotic expansion r/x + F, where r > 0 and F ∈ ℝ((x^{ℝ})) has support lying in (−∞, 0]; in particular, there exist a ∈ ℝ and s > 0 such that f = a + r log x + o(x^{−s}). Hence, e^{if} = e^{ia}x^{ir}e^{io(x^{−s})}. For each t > 0, we have t^{ir} = lim_{x→+∞} e^{if(tx)}/e^{if(x)}, so x^{ir} is definable, hence also the kernel of x^{ir}. □

References

[FHM] A. Fornasiero, P. Hieronymi, and C. Miller, A dichotomy for expansions of the real field, Proc. Amer. Math. Soc. 141 (2013), no. 2, 697–698.

[FM] H. Friedman and C. Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (2001), no. 1, 55–64.

[H1] P. Hieronymi, Defining the set of integers in expansions of the real field by a closed discrete set, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2163–2168.

[H2] , Expansions of subfields of the real field by a discrete set, Fund. Math. 215 (2011), no. 2, 167–175.

[HM] P. Hieronymi and C. Miller, Metric dimensions and tameness in expansions of the real field (2015), preprint, arXiv:1510.00964.

[M1] C. Miller, Tameness in expansions of the real field, Logic Colloquium ’01, 2005, pp. 281– 316.

[M2] , Avoiding the projective hierarchy in expansions of the real field by sequences, Proc. Amer. Math. Soc. 134 (2006), 1483–1493.

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

E-mail address: miller@math.osu.edu
URL: http://https://people.math.osu.edu/miller.1987