    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 . 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 , and an
 $S$
-arithemtic version of a theorem of Dani-Margulis .
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:
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##### References:
  E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Interscience, New York, (1957).  doi: 10.1002/9781118164518. Google Scholar  Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535.  doi: 10.2307/1970210.  Google Scholar  A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math., 83 (1992), 347-372. Google Scholar  Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. Math., 75 (1962), 485-535.  doi: 10.2307/1970210.  Google Scholar  R. Cheung, Integrability Estimates on the Space of $S$-Arithmetic Lattices with Applications to Quadratic Forms, Ph.D. Thesis, Yale University, 2016. Google Scholar  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  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  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  J. Dieudonn, Sur les functions continues $p$-adiques, Bull. Sci. Math., (2), 68 (1994), 79-95. Google Scholar  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  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  G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York, 1959. Google Scholar  G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263-283.   Google Scholar  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  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  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  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  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  A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727.   Google Scholar  V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, (1994). Google Scholar  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  D. Ridout, Indefinite quadratic forms, Mathematika, 5 (1958), 122-124.  doi: 10.1112/S0025579300001443.  Google Scholar  W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2), 17 (1971), 187-253. Google Scholar  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  J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Mathematics, Vol. 1500, Springer-Verlag Berlin Heidelberg New York, (1965).   Google Scholar  J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Études Sci. Publ. Math., 54 (1981), 323-401. Google Scholar  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
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