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An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2020-04-07 , DOI: 10.1112/jlms.12316
Andreas Strömbergsson 1 , Pankaj Vishe 2
Affiliation  

Let G = SL ( 2 , R ) ( R 2 ) k and let Γ be a congruence subgroup of SL ( 2 , Z ) ( Z 2 ) k . We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ G which project to pieces of closed horocycles in SL ( 2 , Z ) SL ( 2 , R ) . As an application, we prove an effective quantitative Oppenheim‐type result for the quadratic form ( m 1 α ) 2 + ( m 2 β ) 2 ( m 3 α ) 2 ( m 4 β ) 2 , for ( α , β ) R 2 of Diophantine type, following the approach by Marklof [Ann. of Math. 158 (2003) 419–471] using theta sums.

中文翻译:

SL(2,R)⋉(R2)⊕k的有效均值分配结果并应用于不均匀二次型

G = SL 2 [R [R 2 ķ 然后让 Γ 是...的全等子组 SL 2 ž ž 2 ķ 。我们证明了特殊类型的单能轨道在多项式有效的渐近均衡分布结果 Γ G 该项目适用于封闭的摩托车零件 SL 2 ž SL 2 [R 。作为应用,我们证明了二次形式的有效定量Oppenheim型结果 1个 - α 2 + 2 - β 2 - 3 - α 2 - 4 - β 2 ,对于 α β [R 2 遵循Marklof的方法[Diophantine类型] 。数学学士学位。158(2003)419–471]使用theta和。
更新日期:2020-04-07
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