On the dimension drop conjecture for diagonal flows on the space of lattices
On the dimension drop conjecture for diagonal flows on the space of lattices
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DOI：
10.1016/j.aim.2023.109058
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发表时间：
2020-10
影响因子：
1.7
通讯作者：
D. Kleinbock;Shahriar Mirzadeh
中科院分区：
文献类型：
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作者：
D. Kleinbock;Shahriar Mirzadeh
设X= G/Γ，其中G是李群，Γ是G中的格，U是X的开子集，{gt}是G的单参数子群。考虑X中g t-轨道错过U的点的集合;如果流是遍历的，则它的测度为零。证明了这个集合的Hausdorff维数严格小于X的维数。当X是紧的或G是真实的秩为1的单李群时，证明了这个猜想。本文证明了G= SL m+ n（R），Γ= SL m+ n（Z）和gt = diag（en t，.，en t，e-m t，.，e-m t）情形下的猜想，实际上提供了余维数的一个有效估计.证明使用指数混合流与SL m+ n（R）/SL m+ n（Z）上高度函数的积分不等式方法。我们还讨论了一个应用的问题，改善狄利克雷定理的同时丢番图逼近。
Abstract Let X= G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {g t} be a one-parameter subgroup of G. Consider the set of points in X whose g t-orbit misses U; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G= SL m+ n (R), Γ= SL m+ n (Z) and g t= diag (e n t,…, e n t, e− m t,…, e− m t), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SL m+ n (R)/SL m+ n (Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.