Skip to main content
SpringerLink
Log in

Khintchine-type theorems for values of subhomogeneous functions at integer points

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers’ second moment estimates. In this paper, we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) \(f: \mathbb {R}^n \rightarrow \mathbb {R}\), we derive a necessary and sufficient condition on the approximating function \(\psi \) for guaranteeing that a generic element \(f\circ g\) in the G-orbit of f is \(\psi \)-approximable; that is, \(|f\circ g(\mathbf {v})| \le \psi (\Vert \mathbf {v}\Vert )\) for infinitely many \(\mathbf {v}\in \mathbb {Z}^n.\) We also deduce a sufficient condition in the case of uniform approximation. Here G can be any closed subgroup of \(\mathrm {ASL}_n(\mathbb {R})\) satisfying certain axioms that allow for the use of Rogers-type estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. An, J., Guan, L., Kleinbock, D.: Nondense orbits on homogeneous spaces and applications to geometry and number theory. arXiv:2001.05174 [math.NT] preprint (2020). Ergodic Theory Dyn. Syst. (to appear)

  2. Athreya, J. S.: Random affine lattices. In: Geometry, groups and dynamics, Contemp. Math., vol. 639, pp. 169–174. Amer. Math. Soc., Providence, RI (2015)

  3. Athreya, J.S., Margulis, G.A.: Logarithm laws for unipotent flows, I. J. Mod. Dyn. 3(3), 359–378 (2009)

    Article  MathSciNet  Google Scholar 

  4. Athreya, J.S., Margulis, G.A.: Values of random polynomials at integer points. J. Mod. Dyn. 12, 9–16 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bandi, P., Ghosh, A., Han, J.: A generic effective Oppenheim theorem for systems of forms. arXiv:2003.06114 [math.NT] preprint (2020)

  6. Bentkus, V., Götze, F.: Lattice point problems and distribution of values of quadratic forms. Ann. Math. 150(3), 977–1027 (1999)

    Article  MathSciNet  Google Scholar 

  7. Beresnevich, V., Velani, S.: A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. Math. 164(3), 971–992 (2006)

    Article  MathSciNet  Google Scholar 

  8. Bourgain, J.: A quantitative Oppenheim theorem for generic diagonal quadratic forms. Israel J. Math. 215(1), 503–512 (2016)

    Article  MathSciNet  Google Scholar 

  9. Broderick, R., Fishman, L., Kleinbock, D., Reich, A., Weiss, B.: The set of badly approximable vectors is strongly \(C^1\) incompressible. Math. Proc. Camb. Philos. Soc. 153(2), 319–339 (2012)

    Article  Google Scholar 

  10. Buterus, P., Götze, F., Hille, T., Margulis, G. A.: Distribution of values of quadratic forms at integral points. arXiv:2003.06114 [math.NT] preprint (2020)

  11. Durrett, R.T.: Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA (1991)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Einsiedler, M., Ward, T.: Functional Analysis, Spectral Theory, and Applications, Graduate Texts in Mathematics, vol. 276. Springer, Cham (2017)

    Book  Google Scholar 

  14. Fishman, L., Kleinbock, D., Merrill, K., Simmons, D.: Diophantine intrinsic approximation on quadric hypersurfaces. arXiv:1405.7650 [math.NT] preprint (2014). J. Eur.Math. Soc. (to appear)

  15. Ghosh, A., Gorodnik, A., Nevo, A.: Optimal density for values of generic polynomial maps. arXiv:1801.01027 [math.NT] preprint (2018)

  16. Ghosh, A., Kelmer, D.: A quantitative Oppenheim theorem or generic ternary quadratic forms. J. Mod. Dyn. 12, 1–8 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ghosh, A., Kelmer, D., Yu, S.: Effective density for inhomogeneous quadratic forms I: generic forms and fixed shifts. arXiv:1911.04739 [math.NT] preprint (2020)

  18. Ghosh, A., Kelmer, D., Yu, S.: Effective density for inhomogeneous quadratic forms II: fixed forms and generic shifts. arXiv:2001.10990 [math.NT] preprint (2020)

  19. Gorodnik, A.: On an Oppenheim-type conjecture for systems of quadratic forms. Israel J. Math. 140, 125–144 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Han, J.: Rogers’ mean value theorem for S-arithmetic Siegel transform and applications to the geometry of numbers. arXiv:1910.01824 [math.NT] preprint (2019)

  22. Kelmer, D., Yu, S.: The second moment of the Siegel transform in the space of symplectic lattices. Int. Math. Res. Notices (2019). https://doi.org/10.1093/imrn/rnz027

    Article  Google Scholar 

  23. Kelmer, D., Yu, S.: Values of random polynomials in shrinking targets. arXiv:1812.04541 [math.NT] preprint (2018)

  24. Kleinbock, D., Margulis, G.A.: Logarithm laws for flows on homogeneous spaces. Invent. Math. 138(3), 451–494 (1999)

    Article  MathSciNet  Google Scholar 

  25. Kleinbock, D., Rao, A.: A zero-one law for uniform Diophantine approximation in Euclidean norm. arXiv:1910.00126 [math.NT] preprint (2019). Int. Math. Res. Notices (to appear)

  26. Kleinbock, D., Wadleigh, N.: An inhomogeneous Dirichlet theorem via shrinking targets. Compos. Math. 155(7), 1402–1423 (2019)

    Article  MathSciNet  Google Scholar 

  27. Kleinbock, D., Weiss, B.: Values of binary quadratic forms at integer points and Schmidt games. In: Recent Trends in Ergodic Theory and Dynamical Systems (Vadodara, 2012), pp. 77–92, Contemp. Math., vol. 631. Amer. Math. Soc., Providence, RI (2015)

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

    Article  MathSciNet  Google Scholar 

  29. Margulis, G.A.: Discrete subgroups and Ergodic theory. Number Theory. Trace Formulas and Discrete Groups (Oslo, 1987), pp. 377–398. Academic Press, Boston, MA (1989)

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

    Article  MathSciNet  Google Scholar 

  31. Rogers, C.A.: Mean values over the space of lattices. Acta Math. 94, 249–287 (1955)

    Article  MathSciNet  Google Scholar 

  32. Rogers, C.A.: The number of lattice points in a set. Proc. Lond. Math. Soc. 6, 305–320 (1956)

    Article  MathSciNet  Google Scholar 

  33. Schmidt, W.M.: A metrical theorem in diophantine approximation. Can. J. Math. 12, 619–631 (1960)

    Article  MathSciNet  Google Scholar 

  34. Schmidt, W.M.: A metrical theorem in geometry of numbers. Trans. Am. Math. Soc. 95(3), 516–529 (1960)

    Article  MathSciNet  Google Scholar 

  35. Schmidt, W.M.: On badly approximable numbers and certain games. Trans. Am. Math. Soc. 123, 178–199 (1966)

    Article  MathSciNet  Google Scholar 

  36. Siegel, C.L.: A mean value theorem in geometry of numbers. Ann. Math. (2) 46, 340–347 (1945)

    Article  MathSciNet  Google Scholar 

  37. Sprindžuk, V.G.: Metric Theory of Diophantine Approximations. Wiley, New York (1979)

    Google Scholar 

  38. Strömbergsson, A.: On the probability of a random lattice avoiding a large convex set. Proc. Lond. Math. Soc. (3) 103(6), 950–1006 (2011)

    Article  MathSciNet  Google Scholar 

  39. Waldschmidt, M.: Recent Advances in Diophantine Approximation Number Theory Analysis and Geometry, pp. 659–704. Springer, New York (2012)

    MATH  Google Scholar 

Download references

Acknowledgements

The first-named author is immensely grateful to Gregory Margulis for a multitude of conversations on the subject of the Oppenheim Conjecture and related topics. Thanks are also due to Jayadev Athreya, Anish Ghosh, Alex Gorodnik, Jiyoung Han, Dubi Kelmer, Dave Morris, and Amos Nevo for stimulating discussions, and to the anonymous referee for several useful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02454-9110, USA

    Dmitry Kleinbock & Mishel Skenderi

Authors
  1. Dmitry Kleinbock
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mishel Skenderi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Kleinbock.

Additional information

Communicated by Adrian Constantin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

D. K. has been supported by NSF Grants DMS-1600814 and DMS-1900560.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kleinbock, D., Skenderi, M. Khintchine-type theorems for values of subhomogeneous functions at integer points. Monatsh Math 194, 523–554 (2021). https://doi.org/10.1007/s00605-020-01498-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-020-01498-1

Keywords

Mathematics Subject Classification

Search

Navigation