Selecta Mathematica

, Volume 24, Issue 2, pp 1633–1675 | Cite as

Diophantine approximations on definable sets

Article
  • 32 Downloads

Abstract

Consider the vanishing locus of a real analytic function on \({{\mathbb {R}}}^n\) restricted to \([0,1]^n\). We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structures which give a general framework to work with sets mentioned above. It complements the theorem of Pila–Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We focus our attention on polynomially bounded o-minimal structures, allow algebraic points of bounded degree, and provide an estimate that is uniform over some families of definable sets. We apply these results to study fixed length sums of roots of unity that are small in modulus.

Mathematics Subject Classification

Primary 11J83 Secondary 03C64 11G50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author is indebted to important suggestions made by Jonathan Pila at an early stage of this work and to Felipe Voloch for pointing out a possible connection to small sums of roots of unity. He is grateful to Victor Beresnevich, David Masser, and Gerry Myerson for comments. He thanks Margaret Thomas and Alex Wilkie for their talks given in Manchester in 2015 and 2013, respectively. He also thanks the Institute for Advanced Study in Princeton, where this work was initiated at the end of 2013, for its hospitality. While there, he was supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. 1.
    Ax, J.: On Schanuel’s conjectures. Ann. Math. 2(93), 252–268 (1971)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beresnevich, V., Vaughan, R.C., Velani, S., Zorin, E.: Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. Int. Math. Res. Not. IMRN 10, 2885–2908 (2017)MathSciNetGoogle Scholar
  3. 3.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36. Springer, Berlin (1998)Google Scholar
  4. 4.
    Bombieri, E., Gubler, W.: Heights in Diophantine Geometry. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  5. 5.
    Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59(2), 337–357 (1989)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Graham, R.L., Sloane, N.J.A.: Anti-Hadamard matrices. Linear Algebra Appl. 62, 113–137 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (2005)MATHGoogle Scholar
  8. 8.
    Huxley, M.N.: The rational points close to a curve. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21(3), 357–375 (1994)MathSciNetMATHGoogle Scholar
  9. 9.
    Huxley, M.N.: The rational points close to a curve. II. Acta Arith. 93(3), 201–219 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jarník, V.: Über die Gitterpunkte auf konvexen Kurven. Math. Z. 24(1), 500–518 (1926)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Konyagin, S.V., Lev, V.F.: On the distribution of exponential sums. Integers A1, 11 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Miller, C.: Expansions of the real field with power functions. Ann. Pure Appl. Log. 68(1), 79–94 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Miller, C.: Exponentiation is hard to avoid. Proc. Am. Math. Soc. 122(1), 257–259 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Minc, H., Sathre, L.: Some inequalities involving \((r!)^{1/r}\). Proc. Edinb. Math. Soc. (2) 14, 41–46 (1964/1965)Google Scholar
  15. 15.
    Myerson, G.: Unsolved problems: how small can a sum of roots of unity be? Am. Math. Mon. 93(6), 457–459 (1986)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pila, J.: Integer points on the dilation of a subanalytic surface. Q. J. Math. 55(2), 207–223 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pila, J.: O-minimality and the André-Oort conjecture for \({\mathbb{C}}^n\). Ann. Math. 173, 1779–1840 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pila, J., Wilkie, A.J.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pila, J., Zannier, U.: Rational points in periodic analytic sets and the Manin–Mumford conjecture. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(2), 149–162 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    van den Dries, L.: A generalization of the Tarski-Seidenberg theorem, and some nondefinability results. Bull. Am. Math. Soc. (N.S.) 15(2), 189–193 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    van den Dries, L.: Tame Topology and O-Minimal Structures, London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  22. 22.
    van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84(2), 497–540 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wilkie, A.J.: Covering definable open sets by open cells. In: Edmundo, M., Richardson, D., Wilkie, A.J. (eds.) Proceedings of the RAAG Summer School Lisbon 2003: O-Minimal Structures. Lecture Notes in Real Algebraic and Analytic Geometry (2005)Google Scholar
  24. 24.
    Wilkie, A.J.: Rational points on definable sets. In: Jones, G.O., Wilkie, A.J. (eds.) O-Minimality and Diophantine Geometry. London Mathematical Society Lecture Note Series, vol. 421, pp. 41–65. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland

Personalised recommendations