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arxiv: 2503.14007 · v1 · submitted 2025-03-18 · 🧮 math.DS

Bounded Geodesics on Locally Symmetric Spaces

Lifan Guan , Chengyang Wu This is my paper

Pith reviewed 2026-05-22 23:55 UTC · model grok-4.3

classification 🧮 math.DS
keywords bounded geodesicslocally symmetric spaceshyperplane absolute winningSL(3,R)geodesic rayshomogeneous dynamics
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The pith

The set of directions for bounded geodesic rays from any point y in Y is hyperplane absolute winning.

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

The paper proves that in the locally symmetric space Y built from SL_3(R) modulo a lattice Γ commensurable with SL_3(Z), every starting point y has a large set of directions where the forward geodesic ray stays inside Y forever. The largeness is expressed by the hyperplane absolute winning property, a condition from dynamical games that is stronger than positive measure or full dimension. A reader would care because it gives a uniform, quantitative description of how common bounded behavior is among geodesics in these spaces, rather than leaving it as an existence or density statement. The proof applies the same way at every y and uses the geometry induced by the invariant metric on the space.

Core claim

Let Γ be a torsion-free subgroup of SL_3(R) commensurable with SL_3(Z), and Y = SO_3(R)backslash SL_3(R)/Γ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

What carries the argument

The hyperplane absolute winning property of the set of directions whose geodesic rays remain bounded in Y.

If this is right

  • The bounded directions satisfy the winning property uniformly at every point y in Y.
  • The result applies to every such space Y arising from the given class of lattices Γ.
  • The winning property is inherited by the geodesic flow on the homogeneous space SL_3(R)/Γ.

Where Pith is reading between the lines

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

  • The same conclusion may hold for other higher-rank groups if the proof adapts to their root systems.
  • The winning property could be combined with homogeneous dynamics to obtain dimension estimates for the set of bounded geodesics.
  • The result supplies a model case for studying bounded orbits under the geodesic flow in other locally symmetric spaces.

Load-bearing premise

Γ must be torsion-free and commensurable with SL_3(Z) so that Y is a well-defined locally symmetric space.

What would settle it

A concrete counterexample would be an explicit point y together with a direction whose ray is bounded yet the collection of all such directions fails to satisfy the hyperplane absolute winning condition under the standard definition of the game.

read the original abstract

Let $\Gamma$ be a torsion-free subgroup of $SL_3(R)$ commensurable with $SL_3(Z)$, and $Y=SO_3(R)\backslash SL_3(R)/\Gamma$ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

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.

Referee Report

0 major / 0 minor

Summary. The paper claims to prove that for any point y in the locally symmetric space Y = SO_3(R) SL_3(R)/Γ, where Γ is a torsion-free subgroup of SL_3(R) commensurable with SL_3(Z), the set of directions in which the geodesic ray starting from y is bounded in Y is hyperplane absolute winning.

Significance. If established with a complete proof, the result would extend the theory of winning sets and Diophantine approximation to the study of bounded geodesics on higher-rank locally symmetric spaces, providing a precise dynamical characterization in the SL_3 setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report on our manuscript. No specific major comments were raised in the report, so we provide no point-by-point responses below. We remain available to address any questions or concerns the referee may have upon further review.

Circularity Check

0 steps flagged

No circularity: direct proof claim with no reduction to inputs or self-citations

full rationale

The paper states a theorem: for torsion-free Γ commensurable with SL_3(Z), the set of bounded geodesic directions from any y in Y is hyperplane absolute winning. No equations, parameters, or fitted quantities appear in the abstract or description. The claim is a mathematical existence/proof result on a specific class of locally symmetric spaces, not a prediction derived from data or prior self-referential definitions. Absent any visible self-citation chains, ansatzes, or renamings that reduce the result to its own inputs, the derivation (whatever its details) is treated as self-contained against external benchmarks in dynamical systems and homogeneous spaces. This matches the expected non-finding for a pure proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard structure of locally symmetric spaces and the definition of hyperplane absolute winning; no free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption Y carries the natural locally symmetric space structure induced from SL_3(R).
    Invoked in the definition of Y and the geodesics.
  • domain assumption Γ is torsion-free and commensurable with SL_3(Z).
    Required for the space Y to be a manifold without singularities.

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    An, Badziahin-Pollington-Velani’s theorem and Schmidt’s gam e, Bull

    J. An, Badziahin-Pollington-Velani’s theorem and Schmidt’s gam e, Bull. Lond. Math. Soc. 45 (2013), no. 4, 721–733

    work page 2013

  2. [2]

    , 2-dimensional badly approximable vectors and Schmidt’s ga me, Duke Math. J. 165 (2016), no. 2, 267–284

    work page 2016

  3. [3]

    J. An, V. Beresnevich, S. Velani, Badly approximable points on planar curves and winning , Adv. Math. 324 (2018), 148–202

    work page 2018

  4. [4]

    J. An, A. Ghosh, L. Guan, T. Ly, Bounded orbits of diagonalizable flows on finite volume quoti ents of products of SL(2, R), Adv. Math. 354 (2019), article no. 106743, 18 pp

    work page 2019

  5. [5]

    J. An, L. Guan, D. Kleinbock, Bounded orbits of diagonalizable flows on SL3(R)/SL3(Z), Int. Math. Res. Not. IMRN 2015 (2015), no. 24, 13623–13652

    work page 2015

  6. [6]

    Systems, 42 (2022), no

    , Nondense orbits on homogeneous spaces and applications to g eometry and number theory , Ergodic Theory Dynam. Systems, 42 (2022), no. 4, 1327–1372

    work page 2022

  7. [7]

    C. S. Aravinda, Bounded geodesics and Hausdorff dimension , in: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 116, no. 3, pp. 505-511. Cambridge University Press, 1994

    work page 1994

  8. [8]

    C. S. Aravinda, E. Leuzinger, Bounded geodesics in rank-1 locally symmetric spaces , Ergodic Theory and Dynamical Systems 15, no. 5 (1995): 813-820

    work page 1995

  9. [9]

    Badziahin, A

    D. Badziahin, A. Pollington, S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture , Ann. of Math. (2) 174 (2011), no. 3, 1837–1883

    work page 2011

  10. [10]

    Beresnevich, Badly approximable points on manifolds , Inventiones mathematicae 202, no

    V. Beresnevich, Badly approximable points on manifolds , Inventiones mathematicae 202, no. 3 (2015): 1199-1240

    work page 2015

  11. [11]

    Beresnevich, E

    V. Beresnevich, E. Nesharim, L. Yang, Bad(w) is hyperplane absolute winning , Geometric and Func- tional Analysis 31 (2021): 1-33

    work page 2021

  12. [12]

    14 (2022): 2841-2880

    , Winning property of badly approximable points on curves , Duke Mathematical Journal 171, no. 14 (2022): 2841-2880

    work page 2022

  13. [13]

    Broderick, Y

    R. Broderick, Y. Bugeaud, L. Fishman, D. Kleinbock, B. W eiss, Schmidt’s game, fractals, and numbers normal to no base , Math. Res Letters, 17 (2010), no. 2, 307–321

    work page 2010

  14. [14]

    Broderick, L

    R. Broderick, L. Fishman, D. Kleinbock, Schmidt’s game, fractals, and orbits of toral endomorphism s, Ergodic Theory Dynam. Systems 31 (2011), no. 4, 1095–1107

    work page 2011

  15. [15]

    Broderick, L

    R. Broderick, L. Fishman, D. Kleinbock, A. Reich, B. Wei ss, The set of badly approximable vectors is strongly C 1 incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319–339

    work page 2012

  16. [16]

    Broderick, L

    R. Broderick, L. Fishman, D. Simmons, Badly approximable systems of affine forms and incompress- ibility on fractals , J. Number Theory 133, no. 7 (2013), 2186–2205

    work page 2013

  17. [17]

    J. W. S. Cassels, An introduction to the geometry of numbers , Corrected reprint of the 1971 edition, Springer-Verlag, Berlin, 1997

    work page 1971

  18. [18]

    Chaika, Y

    J. Chaika, Y. Cheung, and H. Masur, Winning games for bounded geodesics in Teichm¨ uller discs , J. Mod. Dyn. 7 (2013), no. 3, 395–427

    work page 2013

  19. [19]

    S. G. Dani, Invariant measures of horospherical flows on noncompact hom ogeneous spaces, Inventiones mathematicae 47, no. 2 (1978): 101-138

    work page 1978

  20. [20]

    S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Di ophantine approximation , J. Reine Angew. Math. 359 (1985), 55–89

    work page 1985

  21. [21]

    , Bounded orbits of flows on homogeneous spaces , Comment. Math. Helv. 61 (1986), 636–660

    work page 1986

  22. [22]

    Systems 8 (1988), 523–529

    , On orbits of endomorphisms of tori and the Schmidt game , Ergodic Theory Dynam. Systems 8 (1988), 523–529

    work page 1988

  23. [23]

    Fishman, D

    L. Fishman, D. Simmons, M. Urba´ nski, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces , vol. 254, no. 1215. American Mathematical Society, 2018

    work page 2018

  24. [24]

    Kleinbock, Flows on homogeneous spaces and Diophantine properties of m atrices, Duke Math

    D. Kleinbock, Flows on homogeneous spaces and Diophantine properties of m atrices, Duke Math. J. 95 (1998), no. 1, 107–124

    work page 1998

  25. [25]

    Kleinbock, G

    D. Kleinbock, G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous sp aces, Amer. Math. Soc. Transl. 171 (1996), 141–172

    work page 1996

  26. [26]

    Kleinbock, B

    D. Kleinbock, B. Weiss, Modified Schmidt games and diophantine approximation with w eights, Advances in Math. 223 (2010), 1276–1298

    work page 2010

  27. [27]

    30 (2004): 1551-1560

    , Bounded geodesics in moduli space , International Mathematics Research Notices 2004, no. 30 (2004): 1551-1560

    work page 2004

  28. [28]

    , Modified Schmidt games and a conjecture of Margulis , J. Mod. Dyn. 7, no. 3 (2013), 429–460

    work page 2013

  29. [29]

    77–92, Contemp

    , Values of binary quadratic forms at integer points and Schmi dt games , in: Recent trends in ergodic theory and dynamical systems (Vadodara, 2012) , pp. 77–92, Contemp. Math. 631, Amer. Math. Soc., Providence, RI, 2015. 22 LIF AN GUAN AND CHENGYANG WU

    work page 2012

  30. [30]

    G. A. Margulis, Dynamical and ergodic properties of subgroup actions on hom ogeneous spaces with applications to number theory , in: Proceedings of the International Congress of Mathemati- cians (Kyoto, 1990) , pp. 193–215, Math. Soc. Japan, Tokyo, 1991

    work page 1990

  31. [31]

    F. I. Mautner, Geodesic flows on symmetric Riemann spaces , Annals of Mathematics 65, no. 3 (1957): 416-431

    work page 1957

  32. [32]

    C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approxi mation, Geom. Funct. Anal. 20 (2010), no. 3, 726–740

    work page 2010

  33. [33]

    Nesharim, D

    E. Nesharim, D. Simmons, Bad(s, t) is hyperplane absolute winning , Acta Arith. 164 (2014), no. 2, 145–152

    work page 2014

  34. [34]

    Pollington, S

    A. Pollington, S. Velani, On simultaneously badly approximable numbers , J. London Math. Soc. (2) 66 (2002), no. 1, 29–40

    work page 2002

  35. [35]

    W. M. Schmidt, A metrical theorem in diophantine approximation , Canad. J. Math. 12 (1960), 619– 631. [36] , On badly approximable numbers and certain games , Trans. Amer. Math. Soc. 123 (1966), 178–199

    work page 1960

  36. [36]

    Number Theory 1 (1969), 139–154

    , Badly approximable systems of linear forms , J. Number Theory 1 (1969), 139–154

    work page 1969

  37. [37]

    271–287, Progr

    , Open problems in Diophantine approximation , in: Diophantine approximations and transcendental numbers (Luminy, 1982) , pp. 271–287, Progr. Math. 31, Birkh¨ auser, Boston, 1983

    work page 1982

  38. [38]

    Schroeder, Bounded geodesics in manifolds of negative curvature , Mathematische Zeitschrift 235, no

    V. Schroeder, Bounded geodesics in manifolds of negative curvature , Mathematische Zeitschrift 235, no. 4 (2000): 817-828. Institute for Theoretical Sciences, School of Science, Wes tlake University, 600 Dunyu Road, Sandun town, Xihu district, Hangzhou 310030, Zhejian g Province, China. Email address : guanlifan@westlake.edu.cn School of Mathematical Scie...

    work page 2000