论文标题
在尺寸下降的猜想上，在晶格空间上的对角线流动
On the dimension drop conjecture for diagonal flows on the space of lattices
论文作者
论文摘要
让$ x = g/γ$，其中$ g $是一个谎言组，$γ$是$ g $的晶格，让$ u $是$ x $的开放子集，让$ \ {g_t \} $为$ g $的单参数子组。考虑$ x $中的一组积分，其$ g_t $ -orbit会错过$ u $;如果流量为ergodic，它的测量为零。已经猜想这套集合的尺寸严格小于$ x $的尺寸。当$ x $紧凑或$ g $是一个真正的排名$ 1 $的谎言群体时，已证明了这一猜想。在本文中，我们证明了这种猜想的情况，$ g = \ textrm {sl} _ {m+n}（\ Mathbb {r}）$，$γ= \ textrm {sl} _ {m+n} e^{nt}，e^{ - mt}，\ dots，e^{ - mt}）$，实际上为编辑提供了有效的估计。该证明使用流量的指数混合与$ \ textrm {sl} _ {m+n}（\ Mathbb {r}）/\ textrm {sl} _ {m+n}（m+n}（\ Mathbb {Z}）$的指数混合与高度功能的积分不等式方法的指数混合。我们还讨论了在同时进行双苯胺近似中改善Dirichlet定理的问题的应用。
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 has been 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 $G=\textrm{SL}_{m+n}(\mathbb{R})$, $Γ=\textrm{SL}_{m+n}(\mathbb{Z})$ and $g_t=\textrm{diag} (e^{nt}, \dots, e^{nt},e^{-mt}, \dots, e^{-mt})$, 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 $\textrm{SL}_{m+n}(\mathbb{R})/\textrm{SL}_{m+n}(\mathbb{Z})$. We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.
