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Lifan Guan

Publications and source records attributed to Lifan Guan.

12 recordsLinked to original sources
arXiv2026

Singular points for cone actions on the product of certain homogeneous spaces

In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/\Gamma=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\mathbb{Z})^s$, and let $C$ be a cone in the positive Weyl chamber with angular aperture $\epsilon>0$. Then the Hausdorff dimension of the set of points with essential divergent orbits under $C$ satisfies that when $\epsilon\in (0,\frac{1}{64})$, $$ 3s-\frac{1}{2}-4(s-1)\epsilon \leq \dim D^e(C, G/\Gamma)\leq 3s-\frac{1}{2}-\frac{1}{3}\epsilon. $$ This extends the previous result of An--Guan--Marnat--Shi \cite{AGMS} to higher-dimensional cone actions.

math.DS↗
arXiv2025

Bounded Geodesics on Locally Symmetric Spaces

Let $Γ$ be a torsion-free subgroup of $SL_3(R)$ commensurable with $SL_3(Z)$, and $Y=SO_3(R)\backslash SL_3(R)/Γ$ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

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arXiv2021

Nondense orbits on homogeneous spaces and applications to geometry and number theory

Let $G$ be a Lie group, $Γ\subset G$ a discrete subgroup, $X=G/Γ$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories avoiding $Z$ is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on $X$. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.

math.DS↗
arXiv2020

Dirichlet is not just Bad and Singular

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport and Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dircihlet improvable points.

math.NT↗
arXiv2020

Boundary and Eisenstein Cohomology of $G_2(\mathbb{Z})$

In this article, Eisenstein cohomology of the arithmetic group $G_2(\mathbb{Z})$ with coefficients in any finite dimensional highest weight irreducible representation has been determined. We accomplish this by studying the cohomology of the boundary of the Borel-Serre compactification.

math.NT↗
arXiv2020

Divergent trajectories on products of homogeneous spaces

In this paper, we determine the Hausdorff dimension of the set of points with divergent trajectories on the product of certain homogeneous spaces. The flow is allowed to be weighted with respect to the factors in the product space. The result is derived from its counterpart in Diophantine approximation. In doing this, we introduce a notion of jointly singular matrix tuples, and extend the dimension formula for singular matrices to such matrix tuples.

math.DS↗
arXiv2019

Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $SL_2(\mathbb{R})$

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let $G := SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) $ and $Γ$ be a lattice in $G$. We show that the set of points on $G/Γ$ whose forward orbits under a one parameter Ad-semisimple subsemigroup of $G$ are bounded, form a hyperplane absolute winning set.

math.DS↗
arXiv2019

Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents

We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.

math.NT↗
arXiv2018

Hausdorff dimension of divergent trajectories on homogeneous space

For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung.

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