[#P2868] An o-minimal function faster than every finite exponential iterate
Problem. Does there exist an o-minimal expansion \(\mathcal R\) of the ordered real field and an \(\mathcal R\)-definable function \(f:\mathbb R\to\mathbb R\) such that, for every integer \(m\ge1\), there is \(a_m\in\mathbb R\) with \(f(x)>\exp^{\circ m}(x)\) for all \(x>a_m\)?
1Context
The question tests the possible growth scale of tame real geometry. Candidate structures, quantifier-elimination fragments, and Hardy-field comparisons can survive as useful artifacts even when they establish only a fixed growth ceiling.
2Problem setup
Definition 1 (An expansion of the ordered real field adds named functions or relations to \((\mathbb R,<,+,\cdot)\). An expansion of the ordered real field adds named functions or relations to \((\mathbb R,<,+,\cdot)\).
Definition 2 (The expansion). The expansion is o-minimal when every definable subset of \(\mathbb R\) is a finite union of points and open intervals.
Definition 3 (A definable function). A definable function is one whose graph is definable in \(\mathcal R\), with parameters allowed.
Definition 4 (The notation \(\exp^{\circ m}\). The notation \(\exp^{\circ m}\) means the \(m\)-fold iterate of the usual real exponential function.
Remark 1. The question tests the possible growth scale of tame real geometry. Candidate structures, quantifier-elimination fragments, and Hardy-field comparisons can survive as useful artifacts even when they establish only a fixed growth ceiling.
3What counts as a solution
- Construct an explicit o-minimal expansion \(\mathcal R\) and one definable function \(f\), then prove the displayed eventual domination for every positive integer \(m\).
- Alternatively, prove that every unary function definable in every o-minimal expansion of the ordered real field is eventually bounded by some finite iterate of the ordinary exponential.
1Status
Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate. Construct an explicit o-minimal expansion \(\mathcal R\) and one definable function \(f\), then prove the displayed eventual domination for every positive integer \(m\).[1]
1Packet records
Recent contributions
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Notes and companion material
Original intake status. UNKNOWN as of 2026-07-31. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.
- On 2026-07-27 the MathOverflow answer, its 2025 update, and all comments were checked. The update continues to present existence of a non-exponentially-bounded o-minimal structure as open.
- The structure \(\mathbb R_{\mathrm{an},\exp}\) is o-minimal and exponential in character, yet each one of its definable functions is bounded by a suitable fixed finite iterate in the relevant theorems.
- Miller's growth dichotomy explains why adding a sufficiently fast definable function can force definability of the ordinary exponential. Defining exponential still falls short of defining one function above every finite iterate.
- The usual exponential in the comparison is an external fixed real function; the statement does not assume it is named in the language. If the growth dichotomy applies, it may become definable as a consequence.
- Trap: for each \(m\), choosing a different definable function \(f_m=\exp^{\circ(m+1)}\) does not supply the single function \(f\) required here.
Recorded example 1. For a fixed \(r\), the function \(\exp^{\circ r}(x)\) dominates lower iterates but fails the condition when \(m>r\).
How the 2 records connect
ProblemAn o-minimal function faster than every finite exponential iterate
2See also
How to cite
TheoremDB contributors, “An o-minimal function faster than every finite exponential iterate,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/o-minimal-super-iterated-exponentialThis page as plain text: o-minimal-super-iterated-exponential.md
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. An o-minimal expansion with functions of growth higher than iterated exponentials, MathOverflow question 121010. Original CC0 quantified formulation written after reading the answer, its 2025 update, all comments, and standard exponential-boundedness references. Original CC0 quantified formulation written after reading the answer, its 2025 update, all comments, and standard exponential-boundedness references. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For An o-minimal function faster than every finite exponential iterate: UNKNOWN as of 2026-07-27. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.Source named by the research packet.
- Lou van den Dries and Chris Miller, “Geometric Categories and o-Minimal Structures,” Duke Mathematical Journal 84(2) (1996), 497-540. DOI 10.1215/S0012-7094-96-08416-1; revised February 20, 2001. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.Also cited at Sections 1 through 4, especially the o-minimal structure and growth dichotomy results.Source used to assess the problem's recorded status.For An o-minimal function faster than every finite exponential iterate, this source supplies the o-minimal framework and growth results used to delimit the super-iterated-exponential target.
- Lou Van Den Dries and Patrick Speissegger, “The Field of Reals with Multisummable Series and the Exponential Function”. Proceedings of the London Mathematical Society 81(3) (2000), 513-565. DOI 10.1112/S0024611500012648. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.UNKNOWN as of 2026-07-27. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.Also cited at Full journal article relevant to An o-minimal function faster than every finite exponential iterate.Source used to assess the problem's recorded status.For An o-minimal function faster than every finite exponential iterate: UNKNOWN as of 2026-07-27. The source answer, updated in 2025, reports that no non-exponentially-bounded o-minimal expansion is known. The dated search found examples with fixed iterated-exponential growth, but none with one definable function dominating every finite iterate.
Original self-contained restatement motivated by MathOverflow question 121010; domination of each finite iterate is quantified.