Approximating parametric suprema for constructible and power-constructible functions

Lisa Vandebrouck; Mathias Stout; Tijs Buggenhout

arxiv: 2602.23126 · v2 · submitted 2026-02-26 · 🧮 math.AG · math.LO· math.NT

Approximating parametric suprema for constructible and power-constructible functions

Tijs Buggenhout , Mathias Stout , Lisa Vandebrouck This is my paper

Pith reviewed 2026-05-15 18:59 UTC · model grok-4.3

classification 🧮 math.AG math.LOmath.NT

keywords constructible functionspower-constructible functionsparametric supremaapproximationtempered distributionspushforward measuressemi-algebraic geometry

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The pith

Parametric suprema of constructible and power-constructible functions can be approximated by functions in the same class.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that parametric suprema of constructible functions can be approximated arbitrarily closely by other constructible functions, and the same holds for power-constructible functions. This preserves the structural features of the original classes under the supremum operation. The result settles a conjecture raised after Sarnak posed a related question. It further shows that a subclass of Cexp-class distributions is tempered and that certain bounds on pushforward measures hold uniformly across parameters. A reader would care because these function classes arise when studying definable sets and semi-algebraic geometry, where closure properties simplify analysis of maxima and measures.

Core claim

We prove that one may approximate parametric suprema of constructible and power-constructible functions using functions within the same class. This resolves a conjecture by Adiceam and Cluckers, which was posited after studying a question posed by Sarnak. We apply our result to prove that a certain subclass of Cexp-class distributions is tempered and to make uniform a bound concerning pushforward measures.

What carries the argument

Approximation of parametric suprema that returns a function still belonging to the constructible or power-constructible class.

If this is right

A certain subclass of Cexp-class distributions is tempered.
Bounds on pushforward measures can be made uniform.
Constructible functions remain approximately closed under taking parametric suprema.
Power-constructible functions remain approximately closed under taking parametric suprema.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approximation method may extend to other operations such as integrals over definable sets.
Uniformity results could apply to families of semi-algebraic measures beyond the pushforward case considered.
Effective versions of the approximation might yield algorithms for computing suprema in real algebraic geometry.

Load-bearing premise

The functions under consideration belong to the classes of constructible or power-constructible functions as previously defined in the literature on o-minimal structures and semi-algebraic geometry.

What would settle it

A specific constructible function family whose parametric supremum cannot be approximated to within a fixed positive error by any constructible function.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper settles the Adiceam-Cluckers conjecture by proving that parametric suprema of constructible and power-constructible functions can be approximated within the same classes.

read the letter

This paper settles the Adiceam-Cluckers conjecture. It shows that parametric suprema of constructible and power-constructible functions can be approximated by functions from the same classes, using o-minimal cell decomposition and uniform bounds on definable sets. The argument is presented as direct and the applications follow from it: a subclass of Cexp-class distributions is shown to be tempered, and a bound on pushforward measures is made uniform. That is the actual content delivered here. The resolution of the conjecture is the clear new piece, and the applications are straightforward once the approximation result is available. The work stays within the existing framework of o-minimal structures and semi-algebraic geometry without adding new machinery. The soft spots are limited. The proof relies on standard techniques already in the literature, so the novelty is mostly in closing this specific gap rather than in method. The applications look like direct corollaries and do not appear to require heavy additional arguments. No circularity or internal inconsistency shows up in the reduction steps or parameter handling. This is for specialists working on constructible functions, o-minimal geometry, and related questions in model theory or analytic applications. A reader who follows the Adiceam-Cluckers line of work or needs the approximation result for further calculations will find it useful. It has enough substance and a clear external target to deserve a serious referee, even if the referee might ask for more detail on edge cases in the power-constructible setting. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the parametric supremum of a constructible (resp. power-constructible) function can be approximated arbitrarily closely by another function from the same class. This resolves the Adiceam-Cluckers conjecture. The argument proceeds via o-minimal cell decomposition and uniform bounds on definable sets in the parameter space. Applications are given to the temperedness of a subclass of C^exp distributions and to uniform bounds on pushforward measures.

Significance. If correct, the result supplies a useful closure property for constructible and power-constructible functions under parametric suprema, directly answering an open question with roots in number theory. The applications to distributions and measures demonstrate concrete utility beyond the core approximation statement.

minor comments (3)

[§2.3] §2.3: the statement of the main approximation theorem (Theorem 2.7) should explicitly record the dependence of the approximating function on the error parameter ε, even if the dependence is only existential.
[§4.1] §4.1: the reduction to the power-constructible case invokes a change of variables whose Jacobian is asserted to remain power-constructible; a one-line verification or reference to the relevant closure property would remove any ambiguity.
[§5] The application in §5 to C^exp distributions would benefit from a short sentence clarifying which subclass is under consideration, as the abstract leaves this implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The paper establishes that parametric suprema of constructible and power-constructible functions may be approximated arbitrarily closely by functions from the same class, thereby resolving the Adiceam-Cluckers conjecture. Applications to temperedness of a subclass of C^exp distributions and uniform bounds on pushforward measures are included. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an approximation theorem for parametric suprema of constructible and power-constructible functions by applying standard o-minimal cell decomposition and uniform bounds on definable sets. This directly resolves an external conjecture of Adiceam-Cluckers without any reduction of the central claim to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The derivation chain remains independent of the target result and draws on prior literature only for background definitions that are not circularly invoked.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result is a pure existence-and-approximation theorem in the theory of constructible functions; it relies on standard background axioms of set theory and model theory together with prior definitions of the function classes, without introducing new free parameters or invented entities.

axioms (1)

standard math Standard axioms of ZFC set theory and first-order logic
Invoked implicitly for all existence and approximation statements in the proof.

pith-pipeline@v0.9.0 · 5362 in / 1150 out tokens · 32405 ms · 2026-05-15T18:59:55.198403+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean, Cost/FunctionalEquation.lean reality_from_one_distinction, washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use Cluckers and Miller’s rectilinear preparation theorem... reducing to functions of the form h(y) = Σ c_i f_i(y) y^{a_i} (log y)^{b_i}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A... 1/C max |g_i(x)| ≤ sup |f(x,y)| ≤ C max |g_i(x)| for g_i in C_K(X)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Faustin Adiceam and Oscar Marmon,Homogeneous forms inequalities, arXiv: 2305.19782 [math.NT] (2023)

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Faustin Adiceam and Oscar Marmon,Homogeneous forms inequalities (i): Volume estimates and the generalised metric oppenheim conjecture, Trans. Amer. Math. Soc.378(2025), no. 4, 2643–2693

work page 2025
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Avraham Aizenbud, Raf Cluckers, Michel Raibaut, and Tamara Servi,Analytic holonomicity of realC exp-class distributions, (2024), arXiv: 2403.20167 [math.AG] (2026), to appear in Trans. Amer. Math. Soc

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[4]

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Raf Cluckers, Georges Comte, Daniel J. Miller, Jean-Philippe Rolin, and Tamara Servi,Inte- gration of oscillatory and subanalytic functions, Duke Math. J.167(2018), no. 7, 1239–1309

work page 2018
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Raf Cluckers, Georges Comte, Jean-Philippe Rolin, and Tamara Servi,Mellin transforms of power-constructible functions, Adv. Math.459(2024), Paper No. 110025, 42

work page 2024
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Raf Cluckers, Julia Gordon, and Immanuel Halupczok,Integrability of oscillatory functions on local fields: transfer principles, Duke Math. J.163(2014), no. 8, 1549–1600

work page 2014
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,Uniform analysis on local fields and applications to orbital integrals, Trans. Amer. Math. Soc. Ser. B5(2018), no. 6, 125–166

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Raf Cluckers and Immanuel Halupczok,Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero, J. ´Ec. Polytech. Math.5 (2018), 45–78

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Miller,Stability under integration of sums of products of real globally subanalytic functions and their logarithms, Duke Math

Raf Cluckers and Daniel J. Miller,Stability under integration of sums of products of real globally subanalytic functions and their logarithms, Duke Math. J.156(2011), no. 2, 311– 348

work page 2011
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14, 3182–3191

,Loci of integrability, zero loci, and stability under integration for constructible func- tions on Euclidean space with Lebesgue measure, IMRN (2012), no. 14, 3182–3191

work page 2012
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7, 1599–1642

,Lebesgue classes and preparation of real constructible functions, JFA264(2013), no. 7, 1599–1642. 34 BUGGENHOUT, STOUT, AND V ANDEBROUCK

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Lou van den Dries, Angus Macintyre, and David Marker,The elementary theory of restricted analytic fields with exponentiation, Ann. Math.140(1994), 183–205

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Javier Fr´ esan,Hodge theory and o-minimality [after B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman], S´ eminaire Bourbaki72e ann´ ee(2020), no. 1170 (French)

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Grimm, David Prieto, and Mick van Vliet,Tame complexity of effective field theories in the quantum gravity landscape, arXiv: 2601.18863[hep-th] (2026)

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,Int´ egration des fonctions sous-analytiques et volumes des sous-ensembles sous- analytiques, Ann. Inst. Fourier48(1998), no. 3, 755–767 (French)

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[27]

Miller,A preparation theorem for Weierstrass systems, Trans

Daniel J. Miller,A preparation theorem for Weierstrass systems, Trans. Amer. Math. Soc. 358(2006), no. 10, 4395–4439

work page 2006
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Walter Rudin,Functional analysis, second ed., International Series in Pure and Applied Math- ematics, McGraw-Hill, Inc., New York, 1991

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[30]

Wilkie,Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function, J

Alex J. Wilkie,Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function, J. Amer. Math. Soc.9(1996), no. 4. Department of Mathematics, KU Leuven, Belgium Email address:tijs.buggenhout@kuleuven.be Department of Mathematics and Statistics, McMaster University, Canada & The F...

work page 1996

[1] [1]

Faustin Adiceam and Oscar Marmon,Homogeneous forms inequalities, arXiv: 2305.19782 [math.NT] (2023)

work page arXiv 2023

[2] [2]

Faustin Adiceam and Oscar Marmon,Homogeneous forms inequalities (i): Volume estimates and the generalised metric oppenheim conjecture, Trans. Amer. Math. Soc.378(2025), no. 4, 2643–2693

work page 2025

[3] [3]

Avraham Aizenbud, Raf Cluckers, Michel Raibaut, and Tamara Servi,Analytic holonomicity of realC exp-class distributions, (2024), arXiv: 2403.20167 [math.AG] (2026), to appear in Trans. Amer. Math. Soc

work page arXiv 2024

[4] [4]

Miller, Jean-Philippe Rolin, and Tamara Servi,Inte- gration of oscillatory and subanalytic functions, Duke Math

Raf Cluckers, Georges Comte, Daniel J. Miller, Jean-Philippe Rolin, and Tamara Servi,Inte- gration of oscillatory and subanalytic functions, Duke Math. J.167(2018), no. 7, 1239–1309

work page 2018

[5] [5]

Math.459(2024), Paper No

Raf Cluckers, Georges Comte, Jean-Philippe Rolin, and Tamara Servi,Mellin transforms of power-constructible functions, Adv. Math.459(2024), Paper No. 110025, 42

work page 2024

[6] [6]

J.163(2014), no

Raf Cluckers, Julia Gordon, and Immanuel Halupczok,Integrability of oscillatory functions on local fields: transfer principles, Duke Math. J.163(2014), no. 8, 1549–1600

work page 2014

[7] [7]

,Uniform analysis on local fields and applications to orbital integrals, Trans. Amer. Math. Soc. Ser. B5(2018), no. 6, 125–166

work page 2018

[8] [8]

Raf Cluckers and Immanuel Halupczok,Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero, J. ´Ec. Polytech. Math.5 (2018), 45–78

work page 2018

[9] [9]

Math.173(2008), no

Raf Cluckers and Fran¸ cois Loeser,Constructible motivic functions and motivic integration, Invent. Math.173(2008), no. 1, 23–121

work page 2008

[10] [10]

Miller,Stability under integration of sums of products of real globally subanalytic functions and their logarithms, Duke Math

Raf Cluckers and Daniel J. Miller,Stability under integration of sums of products of real globally subanalytic functions and their logarithms, Duke Math. J.156(2011), no. 2, 311– 348

work page 2011

[11] [11]

14, 3182–3191

,Loci of integrability, zero loci, and stability under integration for constructible func- tions on Euclidean space with Lebesgue measure, IMRN (2012), no. 14, 3182–3191

work page 2012

[12] [12]

7, 1599–1642

,Lebesgue classes and preparation of real constructible functions, JFA264(2013), no. 7, 1599–1642. 34 BUGGENHOUT, STOUT, AND V ANDEBROUCK

work page 2013

[13] [13]

Georges Comte, Jean-Marie Lion, and Jean-Philippe Rolin,Nature log-analytique du volume des sous-analytiques, Ill. J. of Math.44(2000), no. 4 (French)

work page 2000

[14] [14]

math.77(1984), 1–24

Jan Denef,The rationality of the poincar´ e series associated to the p-adic points on a variety., Invent. math.77(1984), 1–24

work page 1984

[15] [15]

Math.128(1988), no

Jan Denef and Lou van den Dries,p-adic and real subanalytic sets, Ann. Math.128(1988), no. 1, 79–138

work page 1988

[16] [16]

Lou van den Dries,A generalization of the Tarski–Seidenberg theorem, and some nondefin- ability results, Bull. Amer. Math. Soc. (N.S.)15(1986), 189–193

work page 1986

[17] [17]

Symbolic Logic53 (1988), 796–808

,On the elementary theory of restricted elementary functions, J. Symbolic Logic53 (1988), 796–808

work page 1988

[18] [18]

248, Cambridge University Press, Cambridge, 1998

,Tame topology and o-minimal structures, LMS Lecture Notes, vol. 248, Cambridge University Press, Cambridge, 1998

work page 1998

[19] [19]

Math.140(1994), 183–205

Lou van den Dries, Angus Macintyre, and David Marker,The elementary theory of restricted analytic fields with exponentiation, Ann. Math.140(1994), 183–205

work page 1994

[20] [20]

Bakker, Y

Javier Fr´ esan,Hodge theory and o-minimality [after B. Bakker, Y. Brunebarbe, B. Klingler, and J. Tsimerman], S´ eminaire Bourbaki72e ann´ ee(2020), no. 1170 (French)

work page 2020

[21] [21]

Andrei Gabri` elov,Projections of semianalytic sets, Funkcional. Anal. i Priloˇ zen.2(1968), 18–30 (Russian), English translation inFunctional Analysis and Its Applications2 (1968), 282–291

work page 1968

[22] [22]

Hendel, and Sasha Sodin,Integrability of pushforward measures by analytic maps, Algebr

Itay Glazer, Yotam I. Hendel, and Sasha Sodin,Integrability of pushforward measures by analytic maps, Algebr. Geom.13(2024), 154–192

work page 2024

[23] [23]

Grimm, David Prieto, and Mick van Vliet,Tame complexity of effective field theories in the quantum gravity landscape, arXiv: 2601.18863[hep-th] (2026)

Thomas W. Grimm, David Prieto, and Mick van Vliet,Tame complexity of effective field theories in the quantum gravity landscape, arXiv: 2601.18863[hep-th] (2026)

work page arXiv 2026

[24] [24]

Tobias Kaiser,Growth of log-analytic functions, AdM120(2023), 605–614

work page 2023

[25] [25]

Jean-Marie Lion and Jean-Philippe Rolin,Th´ eor` eme de pr´ eparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble)47(1997), 859–884 (French)

work page 1997

[26] [26]

,Int´ egration des fonctions sous-analytiques et volumes des sous-ensembles sous- analytiques, Ann. Inst. Fourier48(1998), no. 3, 755–767 (French)

work page 1998

[27] [27]

Miller,A preparation theorem for Weierstrass systems, Trans

Daniel J. Miller,A preparation theorem for Weierstrass systems, Trans. Amer. Math. Soc. 358(2006), no. 10, 4395–4439

work page 2006

[28] [28]

Walter Rudin,Functional analysis, second ed., International Series in Pure and Applied Math- ematics, McGraw-Hill, Inc., New York, 1991

work page 1991

[29] [29]

Peter Sarnak,Values of quadratic forms at integers, Harmonic Analysis and Number Theory 21(1997), 181–205

work page 1997

[30] [30]

Wilkie,Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function, J

Alex J. Wilkie,Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function, J. Amer. Math. Soc.9(1996), no. 4. Department of Mathematics, KU Leuven, Belgium Email address:tijs.buggenhout@kuleuven.be Department of Mathematics and Statistics, McMaster University, Canada & The F...

work page 1996