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On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form

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Abstract

We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in 3 variables using dynamics on the space of affine lattices.

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References

  1. Bekka, M.B., Mayer, M.: Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces. London Mathematical Society Lecture Note Series, p 269

  2. Borel, A.: Linear Algebraic Groups, Graduate Texts in Mathematics Series, vol. 126. Springer, New York (1991)

    Book  Google Scholar 

  3. Borel, A.: Values of indefinite quadratic forms at integer point and flows on spaces of lattices. Bull. Am. Math. Soc. 32, 184–204 (1995)

    Article  Google Scholar 

  4. Bourgain, J.: A quantitative Oppenheim theorem for generic diagonal quadratic forms. Israel J. Math. 215, 503–512 (2016)

    Article  MathSciNet  Google Scholar 

  5. Dani, S.G., Margulis, G.A.: Limit distributions of orbits of unipotent flows and values of quadratic forms. In: I. M. Gelfand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, pp. 91–137 (1993)

  6. Dani, S.G.: On values of linear and quadratic forms at integral points. In: Number Theory, in: Trends Math. Birkhäuser, Basel, pp. 107–119 (2000)

  7. Dani, S.G.: Simultaneous Diophantine approximation with quadratic and linear forms. J. Mod. Dyn. 2, 129–139 (2008). (special issue dedicated to G.A. Margulis)

    Article  MathSciNet  Google Scholar 

  8. Dani, S.G., Margulis, G.A.: Orbit closures of generic unipotent flows on homogeneous spaces of \({\mathfrak{sl}}(3, \mathbb{R})\). Math. Ann. 286, 101–128 (1990)

    Article  MathSciNet  Google Scholar 

  9. Eskin, A., Margulis, G.A., Mozes, S.: Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. Math. (2) 147, 93–141 (1998)

    Article  MathSciNet  Google Scholar 

  10. Eskin, A., Margulis, G.A., Mozes, S.: Quadratic forms of signature \((2, 2)\) and eigenvalue spacings on rectangular \(2\)- tori. Ann. Math. (2) 161, 679–725 (2005)

    Article  MathSciNet  Google Scholar 

  11. Ghosh, A., Kelmer, D., Yu, S.: Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts. Int. Math. Res. Notices. https://doi.org/10.1093/imrn/rnaa206

  12. Ghosh, A., Kelmer, D., Yu, S.: Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts. https://arxiv.org/abs/2001.10990

  13. Ghosh, A., Kelmer, D.: A quantitative Oppenheim Theorem for generic ternary quadratic forms. J. Mod. Dyn. 12, 1–8 (2018)

    Article  MathSciNet  Google Scholar 

  14. Ghosh, A., Gorodnik, A., Nevo, A.: Optimal density for values of generic polynomial maps. Am. J. Math. 142(6), 1945–1979 (2020)

    Article  MathSciNet  Google Scholar 

  15. Gorodnik, A.: Oppenheim conjecture for pairs consisting of a quadratic form and a linear form. Trans. Am. Math. Soc. 356(11), 4447–4463 (2004)

    Article  Google Scholar 

  16. Gorodnik, A.: Oppenheim-type conjecture for systems of quadratic forms. Israel J. Math. 356(11), 4447–4463 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Lazar, Y.: On the density of \(S\)-adic integers near some projective \(G\)-varieties. https://arxiv.org/abs/1904.10609

  18. Lazar, Y.: Values of pairs involving one quadratic form and one linear form at \(S\)-integral points. J. Numb. Theory 181, 200–217 (2017)

    Article  MathSciNet  Google Scholar 

  19. Lindenstrauss, E., Margulis, G.: Effective estimates on indefinite ternary forms. Israel J. Math. 203(1), 445–499 (2014)

    Article  MathSciNet  Google Scholar 

  20. Margulis, G.: Discrete Subgroups and Ergodic Theory, Number theory, trace formulas and discrete groups, (Oslo, 1987), 377–398. Academic Press, Boston (1989)

    Google Scholar 

  21. Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, vol. 17. Springer, Berlin (1991)

    Book  Google Scholar 

  22. Margulis, G.A.: Oppenheim Conjecture, pp. 272–327. Fields Medalists’ Lectures. World Sci. Publishing, River Edge (1997)

    Google Scholar 

  23. Margulis, G., Mohammadi, A.: Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. Duke Math. J. 158(1), 121–160 (2011)

    Article  MathSciNet  Google Scholar 

  24. Marklof, J.: Pair correlation densities of inhomogeneous quadratic forms. Ann. Math. (2) 158, 419–471 (2003)

    Article  MathSciNet  Google Scholar 

  25. Morris, D.W.: Ratner’s theorems on unipotent flows, Chicago Lectures in Mathematics (2005)

  26. Raghunathan, M.S.: Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge Series, vol. 68. Springer, Berlin (1972)

    Book  Google Scholar 

  27. Ratner, M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, 235–280 (1991)

    Article  MathSciNet  Google Scholar 

  28. Sargent, O.: Density of values of linear maps on quadratic surfaces. J. Numb. Theory 143, 363–384 (2014)

    Article  MathSciNet  Google Scholar 

  29. Winternitz, P.: Subalgebras of Lie algebras. Example of \({\mathfrak{sl}}\). CRM Proc. Lecture Notes 34, 215–227 (2004)

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Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, 400005, India

    Prasuna Bandi & Anish Ghosh

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  1. Prasuna Bandi
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  2. Anish Ghosh
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Correspondence to Anish Ghosh.

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A. G. gratefully acknowledges support from a grant from the Indo-French Centre for the Promotion of Advanced Research, a Department of Science and Technology, Government of India Swarnajayanti fellowship and a MATRICS grant from the Science and Engineering Research Board. P. B. and A. G. acknowledge support of the Department of Atomic Energy, Government of India, under project 12-R&D-TFR-5.01-0500. This work received support from a grant from the Infosys foundation.

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Bandi, P., Ghosh, A. On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form. Math. Z. 299, 781–796 (2021). https://doi.org/10.1007/s00209-021-02716-8

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