Duke Mathematical Journal

Optimal strong approximation for quadratic forms

Naser T. Sardari

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a nondegenerate integral quadratic form F(x1,,xd) in d5 variables, we prove an optimal strong approximation theorem. Let Ω be a fixed compact subset of the affine quadric F(x1,,xd)=1 over the real numbers. Take a small ball B of radius 0<r<1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies Nδ,Ω(r1m)4+δ for any δ>0. Finally assume that an integral vector (λ1,,λd) mod m is given. Then we show that there exists an integral solution x=(x1,,xd) of F(x)=N such that xiλimodm and xNB, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if N is odd and Nδ,Ω(r1m)6+ϵ. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square-root cancellation in a particular sum that appears in Remark 6.8, we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4.

Article information

Duke Math. J., Volume 168, Number 10 (2019), 1887-1927.

Received: 6 April 2017
Revised: 23 January 2019
First available in Project Euclid: 20 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

number theory sums of squares quadratic forms applications of the Hardy–Littlewood method


Sardari, Naser T. Optimal strong approximation for quadratic forms. Duke Math. J. 168 (2019), no. 10, 1887--1927. doi:10.1215/00127094-2019-0007. https://projecteuclid.org/euclid.dmj/1558339351

Export citation


  • [1] F. C. Auluck and S. Chowla, The representation of a large number as a sum of “almost equal” squares, Proc. Indian Acad. Sci. Math. Sci. A6 (1937), 81–82.
  • [2] T. Browning, V. Kumaraswamy, and R. Steiner, Twisted linnik implies optimal covering exponent for $s^{3}$, Int. Math. Res. Not. IMRN 2019, no. 1, 140–164.
  • [3] J. Bourgain and Z. Rudnick, Restriction of toral eigenfunctions to hypersurfaces and nodal sets, Geom. Funct. Anal. 22 (2012), no. 4, 878–937.
  • [4] P. Chiu, Covering with Hecke points, J. Number Theory 53 (1995), no. 1, 25–44.
  • [5] D. Daemen, Localized solutions in Waring’s problem: The lower bound, Acta Arith. 142 (2010), no. 2, 129–143.
  • [6] W. Duke, J. Friedlander, and H. Iwaniec, Bounds for automorphic $L$-functions, Invent. Math. 112 (1993), no. 1, 1–8.
  • [7] M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355–366.
  • [8] A. Ghosh, A. Gorodnik, and A. Nevo, Diophantine approximation and automorphic∗ spectrum, Int. Math. Res. Not. IMRN 2013, no. 21, 5002–5058.
  • [9] A. Ghosh, A. Gorodnik, and A. Nevo, “Diophantine approximation exponents on homogeneous varieties” in Recent Trends in Ergodic Theory and Dynamical Systems (Vadodara, 2012), Contemp. Math. 631, Amer. Math. Soc., Providence, 2015, 181–200.
  • [10] A. Ghosh, A. Gorodnik, and A. Nevo, Best possible rates of distribution of dense lattice orbits in homogeneous spaces, J. Reine Angew. Math 745 (2018), 155–188.
  • [11] Glyn Harman, Approximation of real matrices by integral matrices, J. Number Theory 34 (1990), no. 1, 63–81.
  • [12] D. Heath-Brown, A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481 (1996), 149–206.
  • [13] C. Hooley, On the greatest prime factor of a cubic polynomial, J. Reine Angew. Math. 303/304 (1978), 21–50.
  • [14] H. Kim, Functoriality for the exterior square of $\mathrm{GL}_{4}$ and the symmetric fourth of $\mathrm{GL}_{2}$, with appendices by D. Ramakrishnan and by H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183.
  • [15] H. Kloosterman, On the representation of numbers in the form $ax^{2}+by^{2}+cz^{2}+dt^{2}$, Acta Math. 49 (1927), no. 3–4, 407–464.
  • [16] A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277.
  • [17] A. Malyšev, On the Representation of Integers by Positive Quadratic Forms, Tr. Mat. Inst. Steklova 65, Steklov Inst. Math., Moscow, 1962.
  • [18] G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators (in Russian), Problemy Peredachi Informatsii 24 (1988), no. 1, 51–60; English translation in Problems Inform. Transmission 24 (1988), no. 1, 39–46.
  • [19] I. Rivin and N. Sardari, Quantum chaos on random Cayley graphs of ${\mathrm{SL}_{2}[\mathbb{Z}/p\mathbb{Z}]}$, Exp. Math., published online 14 Dec 2017.
  • [20] P. Sarnak, Some Applications of Modular Forms, Cambridge Tracts in Math. 99, Cambridge Univ. Press, Cambridge, 1990.
  • [21] P. Sarnak, Letter to Scott Aaronson and Andy Pollington on the Solovay–Kitaev Theorem, personal communication, February 2015, http://publications.ias.edu/sarnak/paper/2637.
  • [22] P. Sarnak, Letter to Stephen D. Miller and Naser Talebizadeh Sardari on optimal strong approximation by integral points on quadrics, personal communication, August 2015, http://publications.ias.edu/sarnak/paper/2637.
  • [23] N. Sardari, Complexity of strong approximation on the sphere, preprint, arXiv:1703.02709v2 [math.NT].
  • [24] N. Sardari, Diameter of Ramanujan graphs and random Cayley graphs, Combinatorica, published online August 2018.
  • [25] C. Siegel, Lectures on Quadratic Forms, Tata Institute of Fundamental Research, Bombay, 1967.
  • [26] R. Tijdeman, Approximation of real matrices by integral matrices, J. Number Theory 24 (1986), no. 1, 65–69 24(1):65–69, 1986.
  • [27] E. Wright, The representation of a number as a sum of five or more squares, Q. J. Math. 4 (1933), 37–51.
  • [28] E. Wright, The representation of a number as a sum of four almost equal squares, Q. J. Math. 8 (1937), 278–279.