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On hyperbolic corners and unit-area triangles in planar sets of large measure

T0 review · 1 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Measurable sets in [0,R]² avoiding vertices of axis-aligned right triangles of area 1/2 have measure O(R²/(log R)^c) for any c<1/4.

desk verdict Paper gets any c<1/4 log-power bound on avoiding axis-aligned right triangles of area 1/2 via a hyperbolic trilinear smoothing inequality plus scale induction; the fixed-area case reaches c<1/2. read the letter →

arxiv 2605.30033 v1 pith:4Q56D2ZW submitted 2026-05-28 math.CA math.CO

Aleksandar Bulj , Vjekoslav Kovač This is my paper · ORCID
classification math.CAmath.CO
keywords hyperboliccornersunit-areatrianglesmeasureboundsErdősconjecturetrilinearsmoothinginequalityplanarsetsavoidingconfigurationsRieszenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies large measurable sets in the plane that avoid three points forming an upward right triangle with legs parallel to the axes and area exactly 1/2. It shows that any such set inside the square [0,R]² must have area at most a constant times R² divided by a positive power of log R, where the power can be any number less than 1/4. The argument relies on a new hyperbolic form of a trilinear smoothing inequality to control the density of the set. When the avoided triangles are required only to have some fixed area rather than specifically 1/2, the same method yields a stronger saving of any power less than 1/2. A matching-style construction demonstrates that sets of size roughly R log R can still avoid the configuration, so the upper bound is not sharp but improves earlier o(R²) results and gives partial progress toward a conjecture of Erdős asking whether the measure stays bounded.

What carries the argument

Hyperbolic variant of the two-dimensional trilinear smoothing inequality, which bounds integrals over the hyperbolic corner configurations (x,y), (x+t,y), (x,y+1/t).

What would settle it

An explicit set A inside [0,R]² of measure larger than C R² / (log R)^{1/4} that still contains no triple of the form (x,y), (x+t,y), (x,y+1/t), or a counterexample showing the hyperbolic smoothing inequality fails to deliver the required decay.

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Extended reading notes

Core claim

For large R, any measurable A subset of [0,R]² that contains no triple (x,y), (x+t,y), (x,y+1/t) with t>0 satisfies |A| = O_c(R²/(log R)^c) whenever c<1/4. The same conclusion holds with any c<1/2 when A avoids all triangles of one fixed positive area. The proofs use a hyperbolic variant of the two-dimensional trilinear smoothing inequality together with, in the fixed-area case, induction on scales that alternately controls density and Riesz energy.

Load-bearing premise

The hyperbolic variant of the trilinear smoothing inequality must hold with the quantitative bounds needed to close the density estimates.

Editorial extensions

If this is right

  • Avoiding the specific area-1/2 configuration forces the measure to be o(R²) with at least a small logarithmic factor.
  • Avoiding all triangles of one fixed area yields a quantitatively stronger logarithmic upper bound.
  • The fixed-area result supplies partial progress on Erdős's question whether the measure must remain O(1).
  • A separate construction shows that measure Ω(R log R) is achievable while avoiding the area-1/2 configuration.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The induction-on-scales technique that controls both density and Riesz energy may transfer to other geometric-configuration problems that mix additive and multiplicative structure.
  • Closing the gap between the proven upper bound O(R² / (log R)^c) and the lower bound Ω(R log R) would require either a stronger smoothing inequality or a different method.
  • If analogous hyperbolic smoothing inequalities exist in higher dimensions, similar measure bounds could apply to corner-avoiding sets in R^d.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper proves that for large R, measurable sets A ⊆ [0,R]² avoiding the configuration (x,y), (x+t,y), (x,y+1/t) (t>0) satisfy |A| = O_c(R²/(log R)^c) for any c<1/4, complemented by a construction of measure Ω(R log R). For sets avoiding triangles of any fixed area the bound improves to any c<1/2. The proofs rely on a hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality together with an induction-on-scales argument that controls both density and Riesz energy.

Significance. If the hyperbolic smoothing inequality holds with constants permitting the stated iteration, the results give the first polylogarithmic improvements over the trivial bound for the hyperbolic-corner problem and advance Erdős' unit-area triangle question beyond Graham's o(R²) result, reaching nearly R²/(log R)^{1/2} in the fixed-area case.

major comments (1)
  1. [§4–5] The induction-on-scales argument in §4–5 invokes the hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality; the manuscript must establish explicit dependence of the constants on the hyperbolic parameter and on the scale of the tested configuration (x,y),(x+t,y),(x,y+1/t) so that the iteration yields a positive power c<1/4 (or c<1/2) without the smoothing constant deteriorating faster than any fixed power of log R.
minor comments (1)
  1. [abstract] The abstract states the bounds but does not indicate the precise range of t or the hyperbolic angle over which the smoothing inequality is proved; a brief statement of the parameter regime would clarify applicability to the induction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the results, and the specific suggestion regarding constant dependence. We address the major comment below and will incorporate the requested details in the revision.

read point-by-point responses

  1. Referee: [§4–5] The induction-on-scales argument in §4–5 invokes the hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality; the manuscript must establish explicit dependence of the constants on the hyperbolic parameter and on the scale of the tested configuration (x,y),(x+t,y),(x,y+1/t) so that the iteration yields a positive power c<1/4 (or c<1/2) without the smoothing constant deteriorating faster than any fixed power of log R.

    Authors: We agree that explicit control on the constants is required to justify the iteration and obtain the stated exponents. The current manuscript invokes the hyperbolic smoothing inequality (Theorem 3.1) but does not spell out the dependence on the hyperbolic parameter t and the dyadic scale in sufficient detail for the induction. In the revised version we will expand the proof of the smoothing inequality (or add an appendix) to record that the constant is at most C_ε (1 + |log t|)^C (scale)^ε for any ε>0, with C independent of t and the scale. This growth is slow enough that, when fed into the induction-on-scales argument of §§4–5, the accumulated loss remains smaller than any fixed power of log R, yielding the claimed c<1/4 for the area-1/2 case and c<1/2 for the fixed-area case. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; bounds derived from external inequality and induction on scales

full rationale

The paper establishes measure upper bounds for sets avoiding hyperbolic corners or fixed-area triangles by citing and applying a hyperbolic variant of the external Christ-Durcik-Roos trilinear smoothing inequality, then using induction on scales to control density and Riesz energy. No self-definitional reductions, no parameters fitted to data then relabeled as predictions, and no load-bearing self-citations appear in the derivation chain. The cited smoothing result is from independent authors and is treated as an external analytic input rather than derived within the paper. The complementary lower-bound construction is an explicit example, not a tautology. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claims rest on standard properties of Lebesgue measure in the plane together with the asserted validity of a new hyperbolic smoothing inequality; no free parameters or invented entities are introduced.

assumptions (2)
  • standard math Lebesgue measurability of subsets of R² and the standard properties of two-dimensional Lebesgue measure
    Used to define |A| and the forbidden point configurations throughout.
  • domain assumption Existence and applicability of a hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality
    Explicitly identified in the abstract as a key ingredient of the proof.

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Pith. "Pith review of On hyperbolic corners and unit-area triangles in planar sets of large measure." pith.science (2026). https://pith.science/paper/4Q56D2ZW

@misc{pith2026260530033,
  author       = {Pith},
  title        = {Pith review of: On hyperbolic corners and unit-area triangles in planar sets of large measure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Q56D2ZW}},
  note         = {Machine review of arXiv:2605.30033}
}
abstract

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $\Omega(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erd\H{o}s, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

Figures

Figures reproduced from arXiv: 2605.30033 by the authors.

Figure 1
Figure 1. Example for the lower bound. On the one hand, since a band {(x, y) ∈ [0, R] 2 : a ⩽ x + y ⩽ b} has area (b 2 − a 2 )/2 for any 0 ⩽ a < b ⩽ R, the whole set AR has measure |AR| = 1 2 Xm j=1  R − 4j + 1 8j 2 − (R − 4j) 2  = R 8 Xm j=1 1 j − m 2 + 1 128 Xm j=1 1 j 2 = 1 8 R log R + O(R). On the other hand, AR contains no triple of the form (1.1). Namely, if it did contain such a triple for some x, y ∈ R and t > 0, … view at source ↗

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Works this paper leans on

45 extracted references · 35 canonical work pages

  1. [1]

    Sets of lattice points that form no squares.Studia Sci

    Miklós Ajtai and Endre Szemerédi. Sets of lattice points that form no squares.Studia Sci. Math. Hungar., 9:9–11, 1974

    1974

  2. [2]

    On a generalization of the Hadwiger-Nelson problem

    Mohammad Bardestani and Keivan Mallahi-Karai. On a generalization of the Hadwiger-Nelson problem. Israel J. Math., 217(1):313–335, 2017.doi:10.1007/s11856-017-1448-4

    work page doi:10.1007/s11856-017-1448-4 2017

  3. [3]

    Aspects of uniformity in recurrence.Colloq

    Vitaly Bergelson, Bernard Host, Randall McCutcheon, and François Parreau. Aspects of uniformity in recurrence.Colloq. Math., 84/85:549–576, 2000. Dedicated to the memory of Anzelm Iwanik.doi:10. 4064/cm-84/85-2-549-576

  4. [4]

    Polynomial extensions of van der Waerden’s and Szemerédi’s theorems.J

    Vitaly Bergelson and Alexander Leibman. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems.J. Amer. Math. Soc., 9(3):725–753, 1996.doi:10.1090/S0894-0347-96-00194-4

    work page doi:10.1090/s0894-0347-96-00194-4 1996

  5. [5]

    Erdős problems.https://www.erdosproblems.com/

    Thomas Bloom. Erdős problems.https://www.erdosproblems.com/. Accessed: January 25, 2026

    2026

  6. [6]

    A Szemerédi type theorem for sets of positive density inRk.Israel J

    Jean Bourgain. A Szemerédi type theorem for sets of positive density inRk.Israel J. Math., 54(3):307–316, 1986.doi:10.1007/BF02764959

    work page doi:10.1007/bf02764959 1986

  7. [7]

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

    Jean Bourgain. A nonlinear version of Roth’s theorem for sets of positive density in the real line.J. Analyse Math., 50:169–181, 1988.doi:10.1007/BF02796120

    work page doi:10.1007/bf02796120 1988

  8. [8]

    Bruce and Malabika Pramanik

    Benjamin B. Bruce and Malabika Pramanik. Two-point patterns determined by curves.Math. Ann., 393(1):571–615, 2025.doi:10.1007/s00208-025-03254-y

    work page doi:10.1007/s00208-025-03254-y 2025

Show all 45 references
  1. [9]

    A polynomial Roth theorem for corners inR2 and a related bilinear singular integral operator.Math

    Xuezhi Chen and Jingwei Guo. A polynomial Roth theorem for corners inR2 and a related bilinear singular integral operator.Math. Ann., 390(1):255–301, 2024.doi:10.1007/s00208-023-02763-y

    2024 doi

  2. [10]

    Two-point polynomial patterns in subsets of positive density inRn

    Xuezhi Chen and Changxing Miao. Two-point polynomial patterns in subsets of positive density inRn. Int. Math. Res. Not. IMRN, 2024(14):10865–10879, 2024.doi:10.1093/imrn/rnae108

    2024 doi

  3. [11]

    Trilinear smoothing inequalities and a variant of the triangular Hilbert transform.Adv

    Michael Christ, Polona Durcik, and Joris Roos. Trilinear smoothing inequalities and a variant of the triangular Hilbert transform.Adv. Math., 390:Paper No. 107863, 60, 2021.doi:10.1016/j.aim.2021. 107863

    2021 doi

  4. [12]

    A Roth-type theorem for dense subsets ofRd.Bull

    Brian Cook, Ákos Magyar, and Malabika Pramanik. A Roth-type theorem for dense subsets ofRd.Bull. Lond. Math. Soc., 49(4):676–689, 2017.doi:10.1112/blms.12043

    2017 doi

  5. [13]

    Croft, Kenneth J

    Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy.Unsolved problems in geometry. Problem Books in Mathematics. Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. doi:10.1007/978-1-4612-0963-8

    1991 doi

  6. [14]

    Improved estimates for polynomial Roth type theorems in finite fields.J

    Dong Dong, Xiaochun Li, and Will Sawin. Improved estimates for polynomial Roth type theorems in finite fields.J. Anal. Math., 141(2):689–705, 2020.doi:10.1007/s11854-020-0113-8

    2020 doi

  7. [15]

    A Szemerédi-type theorem for subsets of the unit cube.Anal

    Polona Durcik and Vjekoslav Kovač. A Szemerédi-type theorem for subsets of the unit cube.Anal. PDE, 15(2):507–549, 2022.doi:10.2140/apde.2022.15.507

    2022 doi

  8. [16]

    On side lengths of corners in positive density subsets of the Euclidean space.Int

    Polona Durcik, Vjekoslav Kovač, and Luka Rimanić. On side lengths of corners in positive density subsets of the Euclidean space.Int. Math. Res. Not. IMRN, 14(22):6844–6869, 2018.doi:10.1093/imrn/rnx093. 24 A. BULJ AND V. KOV AČ

    2018 doi

  9. [17]

    A strong-type Furstenberg-Sárközy theorem for sets of positive measure.J

    Polona Durcik, Vjekoslav Kovač, and Mario Stipčić. A strong-type Furstenberg-Sárközy theorem for sets of positive measure.J. Geom. Anal., 33(8):Paper No. 255, 16, 2023.doi:10.1007/s12220-023-01309-7

    2023 doi

  10. [18]

    Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space.Real Anal

    Paul Erdős. Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space.Real Anal. Exchange, 4(2):113–138, 1978/79

    1978

  11. [19]

    My Scottish Book ‘Problems’

    Paul Erdős. My Scottish Book ‘Problems’. In R. Daniel Mauldin, editor,The Scottish Book, pages 27–33. Birkhäuser, Boston, 1981

    1981

  12. [20]

    Some combinatorial, geometric and set theoretic problems in measure theory

    Paul Erdős. Some combinatorial, geometric and set theoretic problems in measure theory. In D. Kölzow and D. Maharam-Stone, editors,Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26–July 2, 1983, volume 1089 ofLecture Notes in Mathemat...

    1983 doi

  13. [21]

    Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances

    Alejandra Gaitan and Victor Lie. Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances. Preprint, 2024. URL:https://arxiv.org/abs/2402.03451

    2024

  14. [22]

    On partitions ofEn.J

    Ronald Lewis Graham. On partitions ofEn.J. Combin. Theory Ser. A, 28(1):89–97, 1980.doi:10.1016/ 0097-3165(80)90061-8

    1980

  15. [23]

    A multidimensional Szemerédi theorem in integers

    Jingwei Guo, Changxing Miao, and Guoqing Zhan. A multidimensional Szemerédi theorem in integers. Preprint, 2026. URL:https://arxiv.org/abs/2605.06360

    2026 arXiv

  16. [24]

    Lacey, and Fan Yang

    Rui Han, Michael T. Lacey, and Fan Yang. A polynomial Roth theorem for corners in finite fields.Math- ematika, 67(4):885–896, 2021.doi:10.1112/mtk.12108

    2021 doi

  17. [25]

    Springer, New York, 2011.doi:10.1007/ 978-1-4419-6055-9

    Sigurdur Helgason.Integral geometry and Radon transforms. Springer, New York, 2011.doi:10.1007/ 978-1-4419-6055-9

    2011

  18. [26]

    A short proof on the boundedness of triangular Hilbert transform along curves

    Martin Hsu and Fred Yu-Hsiang Lin. A short proof on the boundedness of triangular Hilbert transform along curves. Preprint, arXiv:2410.15791, 2024. URL:https://arxiv.org/abs/2410.15791

    2024

  19. [27]

    Liu, Shachar Lovett, Anthony Ostuni, and Mehtaab Sawhney

    Michael Jaber, Yang P. Liu, Shachar Lovett, Anthony Ostuni, and Mehtaab Sawhney. Quasipolynomial bounds for the corners theorem. Preprint, 2025. URL:https://arxiv.org/abs/2504.07006

    2025

  20. [28]

    Density theorems for anisotropic point configurations.Canad

    Vjekoslav Kovač. Density theorems for anisotropic point configurations.Canad. J. Math., 74(5):1244–1276, 2022.doi:10.4153/S0008414X21000225

    2022 doi

  21. [29]

    Large copies of large configurations in large sets (extended abstract)

    Vjekoslav Kovač. Large copies of large configurations in large sets (extended abstract). In T. Orponen, P. Shmerkin, and H. Wang, editors,Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory. Workshop report 25/2023 of the conference held at Ob...

    2023 doi

  22. [30]

    Coloring and density theorems for configurations of a given volume.Proc

    Vjekoslav Kovač. Coloring and density theorems for configurations of a given volume.Proc. Lond. Math. Soc. (3), 132(3):Paper No. e70143, 56 pp, 2026.doi:10.1112/plms.70143

    2026 doi

  23. [31]

    Polynomial progressions in topological fields

    Ben Krause, Mariusz Mirek, Sarah Peluse, and James Wright. Polynomial progressions in topological fields. Forum Math. Sigma, 12:Paper No. e106, 51, 2024.doi:10.1017/fms.2024.104

    2024 doi

  24. [32]

    Corners with polynomial side length

    Noah Kravitz, Borys Kuca, and James Leng. Corners with polynomial side length. Preprint, 2024. URL: https://arxiv.org/abs/2407.08637

    2024

  25. [33]

    Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control.Adv

    Borys Kuca. Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control.Adv. Math., 448:Paper No. 109700, 61, 2024.doi:10.1016/j.aim.2024.109700

    2024 doi

  26. [34]

    Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees.Israel J

    Borys Kuca. Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees.Israel J. Math., 259(2):589–620, 2024.doi:10.1007/s11856-023-2551-3

    2024 doi

  27. [35]

    On a continuous Sárközy-type problem.Int

    Borys Kuca, Tuomas Orponen, and Tuomas Sahlsten. On a continuous Sárközy-type problem.Int. Math. Res. Not. IMRN, 2023(13):11291–11315, 2023.doi:10.1093/imrn/rnac168

    2023 doi

  28. [36]

    Improved bounds for szemerédi’s theorem

    James Leng, Ashwin Sah, and Mehtaab Sawhney. Improved bounds for szemerédi’s theorem. Preprint,

  29. [37]

    URL:https://arxiv.org/abs/2402.17995

  30. [38]

    Uniform nonlinear Szemerédi theorem for corners in finite fields

    Zi Li Lim. Uniform nonlinear Szemerédi theorem for corners in finite fields. Preprint, 2025. URL:https: //arxiv.org/abs/2501.04887

    2025

  31. [39]

    Doctoral dissertation, University of Bonn, Bonn, Germany, 2025

    Yu-Hsiang Fred Lin.Singular Brascamp-Lieb Forms and Multilinear Fourier Multipliers with Rough or Oscillatory Multipliers. Doctoral dissertation, University of Bonn, Bonn, Germany, 2025

    2025

  32. [40]

    Product of simplices and sets of positive upper density inRd.Math

    Neil Lyall and Ákos Magyar. Product of simplices and sets of positive upper density inRd.Math. Proc. Cambridge Philos. Soc., 165(1):25–51, 2018.doi:10.1017/S0305004117000184

    2018 doi

  33. [41]

    Distance graphs and sets of positive upper density inR d.Anal

    Neil Lyall and Ákos Magyar. Distance graphs and sets of positive upper density inR d.Anal. PDE, 13(3):685–700, 2020.doi:10.2140/apde.2020.13.685

    2020 doi

  34. [42]

    Weak hypergraph regularity and applications to geometric Ramsey theory

    Neil Lyall and Ákos Magyar. Weak hypergraph regularity and applications to geometric Ramsey theory. Trans. Amer. Math. Soc. Ser. B, 9:160–207, 2022.doi:10.1090/btran/61. ON HYPERBOLIC CORNERS AND UNIT-AREA TRIANGLES 25

    2022 doi

  35. [43]

    R.DanielMauldin.Someproblemsinsettheory, analysisandgeometry.InPaul Erdős and his mathematics, I (Budapest, 1999), volume 11 ofBolyai Soc. Math. Stud., pages 493–506. János Bolyai Math. Soc., Budapest, 2002

    1999

  36. [44]

    Daniel Mauldin

    R. Daniel Mauldin. Some problems and ideas of Erdős in analysis and geometry. InErdős centennial, volume 25 ofBolyai Soc. Math. Stud., pages 365–376. János Bolyai Math. Soc., Budapest, 2013.doi: 10.1007/978-3-642-39286-3\_13

    2013 doi

  37. [45]

    Shkredov

    Ilya D. Shkredov. On a problem of Gowers.Izv. Ross. Akad. Nauk Ser. Mat., 70(2):179–221, 2006.doi: 10.1070/IM2006v070n02ABEH002316. Email address:aleksandar.bulj@math.hr Email address:vjekovac@math.hr Department of Mathematics, F aculty of Science, University of Zagreb, Bijeni...

    2006 doi

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