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Applications of Homogeneous Dynamics to Quadratic Forms
동질 동역학의 이차 형식들로의 응용

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Authors
한지영
Advisor
임선희
Major
자연과학대학 수리과학부
Issue Date
2016-02
Publisher
서울대학교 대학원
Keywords
homogeneous dynamicsJacobi theta sumquantitative Oppenheim conjectureequidstribution of unbounded functions
Description
학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 임선희.
Abstract
We study the geometry of quadratic forms using equidistribution theorems in homogeneous dynamics. First we study the mean square limit of exponential sums associated to a rational ellipsoid of arbitrary center. We obtain a lower bound for arbitrary center and that lower bound turns out to be the upper bound as well for ellipsoids with the center of certain diophantine type(see theorem 1.0.4). This result generalizes a work of Marklof.

The second topic is the quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms. For an arbitrary open set $I$ in $\mathbb Q_S$, we show that the number of $S$-integral vectors of norm at most $T$, whose values of an irrational quadratic form are $Q$ in $I$, is asymptotically $c(Q, I)T^{n-2}$ as $T$ goes to infinity.
This is a generalization of a work of Eskin-Margulis-Mozes for real case.
Language
English
URI
https://hdl.handle.net/10371/121303
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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