On divergent on average trajectories for higher rank actions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909424949395456 |
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| author | Kim, Wooyeon |
| author_facet | Kim, Wooyeon |
| contents | For $d\ge 3$ we first show that the Hausdorff dimension of the set of $A$-divergent on average points in the $(d-1)$-dimensional closed horosphere in the space of $d$-dimensional Euclidean lattices, where $A$ is the group of positive diagonal matrices, is at most $\frac{d-1}{2}$. In particular, this upper bound is sharp for $d=3$.
We apply this to compute the Hausdorff dimension of the set of exceptions to the inhomogeneous uniform version of Littlewood conjecture. We say that a pair $(ξ_1,ξ_2)\in\mathbb{R}^2$ satisfies the inhomogeneous Littlewood conjecture if $$\liminf_{q\to\infty}q\|qξ_1-θ_1\|_{\mathbb{Z}}\|qξ_2-θ_2\|_{\mathbb{Z}}=0$$ for all $(θ_1,θ_2)\in\mathbb{R}^2$, where $\|\cdot\|_\mathbb{Z}$ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs $(ξ_1,ξ_2)\in\mathbb{R}^2$ not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On divergent on average trajectories for higher rank actions Kim, Wooyeon Dynamical Systems Number Theory For $d\ge 3$ we first show that the Hausdorff dimension of the set of $A$-divergent on average points in the $(d-1)$-dimensional closed horosphere in the space of $d$-dimensional Euclidean lattices, where $A$ is the group of positive diagonal matrices, is at most $\frac{d-1}{2}$. In particular, this upper bound is sharp for $d=3$. We apply this to compute the Hausdorff dimension of the set of exceptions to the inhomogeneous uniform version of Littlewood conjecture. We say that a pair $(ξ_1,ξ_2)\in\mathbb{R}^2$ satisfies the inhomogeneous Littlewood conjecture if $$\liminf_{q\to\infty}q\|qξ_1-θ_1\|_{\mathbb{Z}}\|qξ_2-θ_2\|_{\mathbb{Z}}=0$$ for all $(θ_1,θ_2)\in\mathbb{R}^2$, where $\|\cdot\|_\mathbb{Z}$ denotes the distance to the nearest integer. We prove that the Hausdorff dimension of the set of pairs $(ξ_1,ξ_2)\in\mathbb{R}^2$ not satisfying the inhomogeneous Littlewood conjecture is $1$, which is equal to the Hausdorff dimension of the conjectural set of exceptions. |
| title | On divergent on average trajectories for higher rank actions |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2403.16559 |