# P2868: An o-minimal function faster than every finite exponential iterate - ID: `P2868` - Reference: `o-minimal-super-iterated-exponential` - Page: https://theoremdb.org/statements/P2868 - Record maturity: Reviewed problem with recorded work ## 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 \(x>a_m\)? ### Problem setup - **Remark.** An expansion of the ordered real field adds named functions or relations to \((\mathbb R,<,+,\cdot)\). - **Definition.** The expansion is o-minimal when every definable subset of \(\mathbb R\) is a finite union of points and open intervals. - **Remark.** A definable function is one whose graph is definable in \(\mathcal R\), with parameters allowed. - **Definition.** The notation \(\exp^{\circ m}\) means the \(m\)-fold iterate of the usual real exponential function. ### What 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. ## 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. 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](#reference-1) ## Work ### Evidence for the current status **Claim 1 (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 \(m\). 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. A complete resolution must satisfy this condition: Construct an explicit o-minimal expansion \(\mathcal R\) and one definable function \(f\), then prove the displayed eventual domination for every positive integer \(m\). ### Background and intake notes 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. - Original intake status: 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. - 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: For a fixed \(r\), the function \(\exp^{\circ r}(x)\) dominates lower iterates but fails the condition when \(m>r\). ### Open directions - **Route 1** (reported): 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](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `o-minimal-super-iterated-exponential`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. 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. https://mathoverflow.net/questions/121010/is-there-any-o-minimal-expansion-of-the-real-field-with-functions-of-growth-high - Also cited at See dataset.references[0] for the exact external source and locator. - Also cited at Editorial research route recorded 2026-07-31 - forum; reference source; checked 2026-07-31 - Source use: citation_only - 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. 2. 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. Sections 1 through 4, especially the o-minimal structure and growth dichotomy results https://people.math.osu.edu/miller.1987/newg.pdf - website; reference source; checked 2026-07-31 - Source use: citation_only - 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. 3. 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. Full journal article relevant to An o-minimal function faster than every finite exponential iterate. https://doi.org/10.1112/S0024611500012648 - scholarly_publication; reference source; checked 2026-08-01 - Source use: citation_only - 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.