Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

Jiyoung Han

doi:10.3934/dcds.2020359

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2021, Volume 41, Issue 5: 2205-2225. Doi: 10.3934/dcds.2020359

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Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

Jiyoung Han

Research Institute of Mathematics, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul, 08826, South Korea

Received: December 31, 2019

Revised: July 31, 2020

Early access: October 2020

Published: May 2021

This paper is supported by the Samsung Science and Technology Foundation under project No. SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) grant funded by the Korea government under project No. 0409-20200150

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Abstract

The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms $ q $ of rank at least 5, the number of integral vectors $ \mathbf v $ such that $ q( \mathbf v) $ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $ \mathbf v $ such that $ q( \mathbf v) $ is in the same interval.

In rank $ 3 $ or $ 4 $, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms([8, Theorem 2.4]). In this paper, we extend this result to the $ S $-arithmetic version.

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Mathematics Subject Classification: Primary: 22F30, 22E45, 20F65; Secondary: 20E08, 37P55.

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Figure 1. The 3-dimensional hyperbolic space $ {\mathbb{H}}^3 $. The measure of the set in (4) is equal to the Lebesgue measure of the grey area on the top of the sphere

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Figure 2. Apartment $ \mathcal A_0 $ of $ \mathcal{B}_3 $. $ K_p\setminus {\mathrm{SO}}(2x_1x_3-x_2^2) $ is embedded in the inverse image of the blue line

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