# American Institute of Mathematical Sciences

2017, 11: 501-550. doi: 10.3934/jmd.2017020

## Asymptotic distribution of values of isotropic here quadratic forms at S-integral points

 1 Department of Mathematical Sciences, Seoul National University, Kwanak-ro 1, Kwanak-gu, Seoul 08826, Republic of Korea 2 Department of Mathematics, Jacobs University, 28759 Bremen, Germany

Received  June 09, 2016 Revised  August 02, 2017 Published  November 2017

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places
 $S$
over
 ${\mathbb{Q}}$
containing the Archimedean place and excluding the prime
 $2$
, an irrational isotropic form
 ${\mathbf q}$
of rank
 $n\geq 4$
on
 ${\mathbb{Q}}_S$
, a product of
 $p$
 $\mathsf{I}_p$
, and a product
 $\Omega$
of star-shaped sets. We show that unless
 $n=4$
and
 ${\mathbf q}$
is split in at least one place, the number of
 $S$
-integral vectors
 $\mathbf v \in {\mathsf{T}} \Omega$
satisfying simultaneously
 ${\mathbf q}(\mathbf v) \in I_p$
for
 $p \in S$
is asymptotically given by
 $\begin{split}\lambda({\mathbf q}, \Omega) |\,\mathsf{I}\,| \cdot \| {\mathsf{T}} \|^{n-2}\end{split}$
as
 ${\mathsf{T}}$
goes to infinity, where
 $|\,\mathsf{I}\,|$
is the product of Haar measures of the
 $p$
 $I_p$
. The proof uses dynamics of unipotent flows on
 $S$
-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an
 $S$
-arithmetic variant of the
 $\alpha$
-function introduced in [10], and an
 $S$
-arithemtic version of a theorem of Dani-Margulis [7].
Citation: Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020
##### References:
 [1] E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Interscience, New York, (1957). doi: 10.1002/9781118164518. Google Scholar [2] Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535. doi: 10.2307/1970210. Google Scholar [3] A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math., 83 (1992), 347-372. Google Scholar [4] Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535. doi: 10.2307/1970210. Google Scholar [5] R. Cheung, Integrability Estimates on the Space of $S$-Arithmetic Lattices with Applications to Quadratic Forms, Ph.D. Thesis, Yale University, 2016. Google Scholar [6] H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, second edition, with a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, edited and prepared for publication by T. D. Browning, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542893. Google Scholar [7] S. Dani and G. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., 16, Part 1, Amer. Math. Soc., Providence, RI, (1993), 91-137. Google Scholar [8] H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3), 9 (1959), 544-555. doi: 10.1112/plms/s3-9.4.544. Google Scholar [9] J. Dieudonn, Sur les functions continues $p$-adiques, Bull. Sci. Math., (2), 68 (1994), 79-95. Google Scholar [10] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar [11] A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature $(2,2)$ and eignevalue spacings on rectangular $2$-tori, Ann. of Math. (2), 161 (2005), 679-725. doi: 10.4007/annals.2005.161.679. Google Scholar [12] G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York, 1959. Google Scholar [13] G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263-283. Google Scholar [14] M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, in Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 275–290. Google Scholar [15] D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102. Google Scholar [16] E. Landau, Neue Untersuchungen über die Pfeiffer'sche Methode zur Abschätzung von Gitterpunktanzahlen, in Sitzungsber. D. Math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2, Abteilung, Wien, (1915), 469-505. Google Scholar [17] G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, (French summary), C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253. Google Scholar [18] G. A. Margulis and G. Tomanov, Invariant measures of unipotent groups over local fields on homogeneous spaces, Inv. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565. Google Scholar [19] A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727. Google Scholar [20] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, (1994). Google Scholar [21] M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2. Google Scholar [22] D. Ridout, Indefinite quadratic forms, Mathematika, 5 (1958), 122-124. doi: 10.1112/S0025579300001443. Google Scholar [23] W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2), 17 (1971), 187-253. Google Scholar [24] J.-P. Serre, A Course in Arithmetic, translated from the French, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, (1973). Google Scholar [25] J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Mathematics, Vol. 1500, Springer-Verlag Berlin Heidelberg New York, (1965). Google Scholar [26] J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Études Sci. Publ. Math., 54 (1981), 323-401. Google Scholar [27] G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, (1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, (2000), 265-297. Google Scholar

show all references

##### References:
 [1] E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Interscience, New York, (1957). doi: 10.1002/9781118164518. Google Scholar [2] Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535. doi: 10.2307/1970210. Google Scholar [3] A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math., 83 (1992), 347-372. Google Scholar [4] Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535. doi: 10.2307/1970210. Google Scholar [5] R. Cheung, Integrability Estimates on the Space of $S$-Arithmetic Lattices with Applications to Quadratic Forms, Ph.D. Thesis, Yale University, 2016. Google Scholar [6] H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, second edition, with a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, edited and prepared for publication by T. D. Browning, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542893. Google Scholar [7] S. Dani and G. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., 16, Part 1, Amer. Math. Soc., Providence, RI, (1993), 91-137. Google Scholar [8] H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3), 9 (1959), 544-555. doi: 10.1112/plms/s3-9.4.544. Google Scholar [9] J. Dieudonn, Sur les functions continues $p$-adiques, Bull. Sci. Math., (2), 68 (1994), 79-95. Google Scholar [10] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984. Google Scholar [11] A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature $(2,2)$ and eignevalue spacings on rectangular $2$-tori, Ann. of Math. (2), 161 (2005), 679-725. doi: 10.4007/annals.2005.161.679. Google Scholar [12] G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York, 1959. Google Scholar [13] G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263-283. Google Scholar [14] M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, in Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 275–290. Google Scholar [15] D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102. Google Scholar [16] E. Landau, Neue Untersuchungen über die Pfeiffer'sche Methode zur Abschätzung von Gitterpunktanzahlen, in Sitzungsber. D. Math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2, Abteilung, Wien, (1915), 469-505. Google Scholar [17] G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, (French summary), C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253. Google Scholar [18] G. A. Margulis and G. Tomanov, Invariant measures of unipotent groups over local fields on homogeneous spaces, Inv. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565. Google Scholar [19] A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727. Google Scholar [20] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, (1994). Google Scholar [21] M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2. Google Scholar [22] D. Ridout, Indefinite quadratic forms, Mathematika, 5 (1958), 122-124. doi: 10.1112/S0025579300001443. Google Scholar [23] W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2), 17 (1971), 187-253. Google Scholar [24] J.-P. Serre, A Course in Arithmetic, translated from the French, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, (1973). Google Scholar [25] J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Mathematics, Vol. 1500, Springer-Verlag Berlin Heidelberg New York, (1965). Google Scholar [26] J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Études Sci. Publ. Math., 54 (1981), 323-401. Google Scholar [27] G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, (1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, (2000), 265-297. Google Scholar
 [1] Alex Eskin, Gregory Margulis and Shahar Mozes. On a quantitative version of the Oppenheim conjecture. Electronic Research Announcements, 1995, 1: 124-130. [2] Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015 [3] Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541 [4] Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062 [5] Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001 [6] Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082 [7] Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457 [8] Michael Hutchings, Frank Morgan, Manuel Ritore and Antonio Ros. Proof of the double bubble conjecture. Electronic Research Announcements, 2000, 6: 45-49. [9] G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100. [10] Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73. [11] Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 473-496. doi: 10.3934/dcdss.2010.3.473 [12] Joel Hass, Michael Hutchings and Roger Schlafly. The double bubble conjecture. Electronic Research Announcements, 1995, 1: 98-102. [13] Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004 [14] K. H. Kim and F. W. Roush. The Williams conjecture is false for irreducible subshifts. Electronic Research Announcements, 1997, 3: 105-109. [15] Yakov Varshavsky. A proof of a generalization of Deligne's conjecture. Electronic Research Announcements, 2005, 11: 78-88. [16] Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429 [17] Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455 [18] François Lalonde, Egor Shelukhin. Proof of the main conjecture on $g$-areas. Electronic Research Announcements, 2015, 22: 92-102. doi: 10.3934/era.2015.22.92 [19] Petter Branden. Counterexamples to the Neggers-Stanley conjecture. Electronic Research Announcements, 2004, 10: 155-158. [20] Yannick Sire, Christopher D. Sogge, Chengbo Wang. The Strauss conjecture on negatively curved backgrounds. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7081-7099. doi: 10.3934/dcds.2019296

2018 Impact Factor: 0.295