Duke Mathematical Journal

Optimal strong approximation for quadratic forms

Naser T. Sardari

Abstract

For a nondegenerate integral quadratic form $F(x_{1},\dots ,x_{d})$ in $d\geq 5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_{1},\dots ,x_{d})=1$ over the real numbers. Take a small ball $B$ of radius $0\lt r\lt 1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg _{\delta ,\Omega }(r^{-1}m)^{4+\delta }$ for any $\delta \gt 0$. Finally assume that an integral vector $(\lambda _{1},\dots ,\lambda _{d})$ mod $m$ is given. Then we show that there exists an integral solution $\mathbf{x}=(x_{1},\dots ,x_{d})$ of $F(\mathbf{x})=N$ such that $x_{i}\equiv \lambda_{i}\ \mathrm{mod}\ m$ and $\frac{\mathbf{x}}{\sqrt{N}}\in B$, 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 $N\gg _{\delta ,\Omega }(r^{-1}m)^{6+\epsilon }$. 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

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

Dates
Revised: 23 January 2019
First available in Project Euclid: 20 May 2019

https://projecteuclid.org/euclid.dmj/1558339351

Digital Object Identifier
doi:10.1215/00127094-2019-0007

Mathematical Reviews number (MathSciNet)
MR3983294

Citation

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

References

• [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.