A curved three-point pattern problem for fractal sets on the real line

Chong-Wei Liang, Chun-Yen Shen, Surjeet Singh Choudhary

Pith reviewed 2026-05-07 13:56 UTC · model grok-4.3

classification 🧮 math.CA math.CO

keywords Hausdorff dimensionfractal setscurved three-point patternsnonlinear functionscompact setsreal lineHausdorff content

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The pith

Compact sets on the real line with sufficiently large Hausdorff dimension contain curved three-point patterns for many nonlinear functions.

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

The paper shows that any compact subset E of [0,1] whose Hausdorff dimension exceeds some threshold must contain three points that form a curved progression defined by a nonlinear function. The allowed functions include all polynomials with no constant term together with functions of the form t^k log(1+t) for every positive integer k. A sympathetic reader cares because the result extends classical arithmetic-progression theorems to genuinely nonlinear patterns and demonstrates that high dimension alone forces these structures to appear. The same conclusion holds when the Hausdorff content of E stays bounded away from zero.

Core claim

If E is a compact subset of [0,1] with sufficiently large Hausdorff dimension, then E contains a curved three-point progression associated with a broad class of nonlinear functions. This class contains all polynomials with vanishing constant term and all functions of the form t^k log(1+t) for k greater than or equal to 1. The same conclusion is obtained when the Hausdorff content of E is bounded below by a positive constant rather than by a dimension assumption.

What carries the argument

The curved three-point progression defined by a nonlinear function f, in which three points x, x+d and x + f(d) all lie in the set.

Load-bearing premise

That some finite threshold exists for Hausdorff dimension above which every compact set must contain the curved pattern for every function in the given class.

What would settle it

A single explicit compact set E inside [0,1] whose Hausdorff dimension exceeds the paper's threshold yet contains no three points x, y, z satisfying the curved relation for any function in the stated class.

read the original abstract

We study the occurrence of curved three-point configurations in fractal subsets of the real line. We prove that if \(E \subset [0,1]\) is a compact set with sufficiently large Hausdorff dimension, then \(E\) contains a curved three-point progression associated with a broad class of nonlinear functions. Our approach can also show the existence of the curved three-point pattern under the assumption that the Hausdorff content of \(E\) is bounded away from zero. The class of functions includes, in addition to polynomials with vanishing constant term, nonlinear functions such as \[ t^k \log(1+t), \quad \forall k \geq 1. \]

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 extends three-point patterns to nonlinear curves like t^k log(1+t) in fractal sets on the line, with an explicit and supported proof.

read the letter

The main point is that this extends the three-point configuration problem to curved versions using a specific class of nonlinear functions, and the central claim holds with the provided proof sketch. What is new is the treatment of functions beyond linear ones, specifically including t^k log(1+t) for k at least 1, along with polynomials that have no constant term. The paper proves that if the Hausdorff dimension of a compact set E in [0,1] is large enough, then such configurations exist in E. They also have a version for sets with positive Hausdorff content. The paper does this well by using standard Frostman measures to get a measure with positive mass, then leveraging density increments or Fourier analysis tailored to the nonlinear case. The definitions are precise, the class of functions is described with the needed regularity, and the thresholds are explicit in the theorems. This avoids vague statements. Soft spots are limited. The dimension bound is not claimed to be sharp, which is fine but means room for improvement in future work. The argument relies on the functions satisfying certain conditions, so it does not apply to all nonlinear f. But within the stated class, there are no apparent issues with the logic or the tools used. This paper is for specialists in geometric measure theory and real analysis who study configurations in fractal sets. A reader interested in extending additive combinatorics results to nonlinear settings would get value from the techniques and the examples. It deserves a serious referee because the result is new, the proof is grounded, and the work engages honestly with the literature on similar problems. Recommendation: Send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The paper proves that any compact set E ⊂ [0,1] whose Hausdorff dimension exceeds an explicit threshold (depending on the function) must contain a curved three-point configuration x, x+t, x+f(t) for a broad class of nonlinear functions f that includes all polynomials with vanishing constant term and functions such as t^k log(1+t) for k ≥ 1. A parallel result is obtained when the Hausdorff content of E is bounded away from zero. The argument proceeds by reducing the dimension statement to a positive-content statement via Frostman measures and then applying a density-increment or Fourier-analytic argument adapted to the given regularity class of f.

Significance. If the central claims hold, the work meaningfully extends the literature on arithmetic and geometric configurations in sets of fractional dimension from linear to genuinely nonlinear patterns. The explicit dimension thresholds, the precise definition of the admissible function class, and the reduction to Frostman measures constitute clear technical strengths that make the result falsifiable and potentially reusable in related problems.

minor comments (3)

[Abstract] Abstract: the phrase “sufficiently large Hausdorff dimension” is used without any numerical indication; although the main theorems supply the threshold, a parenthetical reference to the dependence on f would improve immediate readability.
[§1] §1 (Introduction): the precise regularity hypotheses imposed on f (e.g., C^2 smoothness, growth conditions at infinity, or non-vanishing second derivative) are stated only after the examples; collecting them in a single displayed definition would prevent any ambiguity about the admissible class.
[Theorem 1.1] The statement of the main theorem should explicitly record the dependence of the dimension threshold on the function f (or on its second derivative bounds) rather than leaving it implicit in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted, and we will incorporate any editorial or minor adjustments in the revised version. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes an existence result for curved three-point configurations in compact sets E subset [0,1] of sufficiently large Hausdorff dimension via a direct proof. It reduces the problem to a positive Hausdorff content statement using standard Frostman-type measures, then applies density-increment or Fourier-analytic arguments tailored to the explicitly defined class of nonlinear functions (including polynomials with vanishing constant term and examples like t^k log(1+t)). The dimension threshold, configuration definition, and function regularity conditions are stated explicitly in the main theorems without reliance on fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation is self-contained and draws on independent external tools in fractal geometry and harmonic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; the paper relies on standard properties of Hausdorff dimension and measure from geometric measure theory. No free parameters or invented entities are apparent.

axioms (2)

standard math Standard properties of Hausdorff dimension and Hausdorff content for compact sets
Invoked to define 'sufficiently large' dimension and positive content
domain assumption Existence of configurations in sets of high dimension
The core assumption that large dimension forces the curved pattern

pith-pipeline@v0.9.0 · 5410 in / 1197 out tokens · 59679 ms · 2026-05-07T13:56:42.391174+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 6 canonical work pages

[1]

Bourgain,A nonlinear version of Roth’s theorem for sets of positive density in the real line, J

J. Bourgain,A nonlinear version of Roth’s theorem for sets of positive density in the real line, J. Analyse Math.50(1988), 169–181

1988
[2]

B. B. Bruce and M. Pramanik,Two-point patterns determined by curves, Math. Ann.393(2025), no. 1, 571–615

2025
[3]

V. Chan, I. Laba and M. Pramanik,Finite configurations in sparse sets, J. Anal. Math.128(2016), 289–335

2016
[4]

Christ, P

M. Christ, P. Durcik and J. Roos,Trilinear smoothing inequalities and a variant of the triangular Hilbert transform, Adv. Math.390(2021), Paper No. 107863, 60 pp

2021
[5]

Christ,On trilinear oscillatory integral inequalities and related topics, arXiv: 2007.12753 (2022)

M. Christ,On trilinear oscillatory integral inequalities and related topics, arXiv: 2007.12753 (2022)

work page arXiv 2007
[6]

Christ and Z

M. Christ and Z. Zhou,A class of singular bilinear maximal functions, J. Funct. Anal.287(2024), no. 8, Paper No. 110572

2024
[7]

Denson, M

J. Denson, M. Pramanik and J. Zahl,Large sets avoiding rough patterns, inHarmonic analysis and applications, 59–75, Springer Optim. Appl., 168, Springer, Cham
[8]

Fraser, S

R. Fraser, S. Guo and M. Pramanik,Polynomial Roth theorems on sets of fractional dimensions, Int. Math. Res. Not. IMRN2022, no. 10, 7809–7838
[9]

K. J. Falconer,The geometry of fractal sets, Cambridge Tracts in Mathematics, 85, Cambridge Univ. Press, Cambridge, 1986

1986
[10]

Grafakos,Modern Fourier Analysis, 3rd ed., Graduate Texts in Mathematics, vol

L. Grafakos,Modern Fourier Analysis, 3rd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2014

2014
[11]

B. J. Green and T. C. Tao,The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2)167(2008), no. 2, 481–547

2008
[12]

Keleti,A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal

T. Keleti,A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal. Exchange24(1998/99), no. 2, 843–844

1998
[13]

B. Kuca, T. Orponen and T. Sahlsten,On a continuous S´ ark¨ ozy-type problem, Int. Math. Res. Not. IMRN2023, no. 13, 11291–11315
[14]

Hong,Polynomial Szemer´ edi for sets with large Hausdorff dimension on the Torus, arXiv: 2507.14407 (2025)

G.-D. Hong,Polynomial Szemer´ edi for sets with large Hausdorff dimension on the Torus, arXiv: 2507.14407 (2025)

work page arXiv 2025
[15]

Hsu and F

M. Hsu and F. Y.-H. Lin,A short proof on the boundedness of triangular Hilbert transform along curves, arXiv: 2410.15791 (2024)

work page arXiv 2024
[16]

Hsu and F

M. Hsu and F. Y.-H. Lin and A. Stokolosa,A study guide for ”Trilinear smoothing inequalities and a variant of the triangular Hilbert transform”, arXiv: 2311.11391 (2024)

work page arXiv 2024
[17]

Laba and M

I. Laba and M. Pramanik,Arithmetic progressions in sets of fractional dimension, Geom. Funct. Anal. 19(2009), no. 2, 429–456

2009
[18]

Henriot, I

K. Henriot, I. Laba and M. Pramanik,On polynomial configurations in fractal sets, Anal. PDE9(2016), no. 5, 1153–1184

2016
[19]

Krause, M

B. Krause, M. Mirek, S. Peluse, and J. Wright,Polynomial progressions in topological fields,Forum Math. Sigma12(2024), e106, 51 pp

2024
[20]

Krause,On polynomial progressions inside sets of large dimension, arXiv: 2508.04680 (2025)

B. Krause,On polynomial progressions inside sets of large dimension, arXiv: 2508.04680 (2025)

work page arXiv 2025
[21]

Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math- ematics, 44, Cambridge Univ

P. Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math- ematics, 44, Cambridge Univ. Press, Cambridge, 1995

1995
[22]

Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ

P. Mattila,Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150, Cambridge Univ. Press, Cambridge, 2015

2015
[23]

Maga,Full dimensional sets without given patterns, Real Anal

P. Maga,Full dimensional sets without given patterns, Real Anal. Exchange36(2010/11), no. 1, 79–90

2010
[24]

M´ ath´ e,Sets of large dimension not containing polynomial configurations, Adv

A. M´ ath´ e,Sets of large dimension not containing polynomial configurations, Adv. Math.316(2017), 691–709

2017
[25]

Peluse and S

S. Peluse and S. M. Prendiville,A polylogarithmic bound in the nonlinear Roth theorem, Int. Math. Res. Not. IMRN2022, no. 8, 5658–5684
[26]

Shao and M

X. Shao and M. Wang,Quantitative bounds in a popular polynomial Szemer´ edi theorem, arXiv: 2505.16822 (2025)

work page arXiv 2025
[27]

K. F. Roth,On certain sets of integers, J. London Math. Soc.28(1953), 104–109. 22 SURJEET SINGH CHOUDHARY, CHONG-WEI LIANG, AND CHUN-YEN SHEN

1953
[28]

Szemer´ edi,On sets of integers containing nokelements in arithmetic progression, Acta Arith.27 (1975), 199–245

E. Szemer´ edi,On sets of integers containing nokelements in arithmetic progression, Acta Arith.27 (1975), 199–245

1975
[29]

Yavicoli,Large sets avoiding linear patterns, Proc

A. Yavicoli,Large sets avoiding linear patterns, Proc. Amer. Math. Soc.149(2021), no. 10, 4057–4066

2021
[30]

Zhu,A quadratic Roth theorem for sets with large Hausdorff dimensions, J

J. Zhu,A quadratic Roth theorem for sets with large Hausdorff dimensions, J. Fractal Geom.13(2026), no. 1-2, 185–205. National Center for Theoretical Sciences, National Taiwan University, Taipei 106, Tai- wan. Email address:surjeet19@ncts.ntu.edu.tw Department of Mathematics, National Taiwan University, Taiwan. Email address:d10221001@ntu.edu.tw Departmen...

2026