ASYMPTOTIC DISTRIBUTION OF VALUES OF ISOTROPIC QUADRATIC FORMS AT S-INTEGRAL POINTS

作者：Han Jiyoung^*; Lim Seonhee; Mallahi Karai Keivan

来源：Journal of Modern Dynamics, 2017, 11(03): 501-550.

DOI：10.3934/jmd.2017020

• Oppenheim conjecture
• homogeneous dynamics

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places S over Q containing the Archimedean place and excluding the prime 2, an irrational isotropic form q of rank n >= 4 on Q(S), a product of p-adic intervals I-p, and a product Omega of star-shaped sets. We show that unless n = 4 and q is split in at least one place, the number of S-integral vectors v is an element of T Omega satisfying simultaneously q(v) is an element of I-p for p is an element of S is asymptotically given by
lambda(q,Omega) vertical bar I vertical bar . parallel to T parallel to(n-2)
as T goes to infinity, where vertical bar I vertical bar is the product of Haar measures of the p-adic intervals 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].

• 出版日期2017