arXiv subjects

Zhiren Wang

Publications and source records attributed to Zhiren Wang.

39 records · Page 3Linked to original sources

arXiv2012

Remarks on Euclidean Minima

The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean spectrum $Spec(K)$, is isolated and attained and can be computed in finite time. We extend Cerri's works by applying recent dynamical results of Lindenstrauss and Wang. In particular, the following facts are proved: (1) For any number field $K$ of unit rank 3 or higher, $M(K)$ is isolated and attained and Cerri's algorithm computes $M(K)$ in finite time. (2) If $K$ is a non-CM field of unit rank 2 or higher, then the computational complexity of $M(K)$ is bounded in terms of the degree, discriminant and regulator of $K$.

math.NT↗

arXiv2012

Topological Self-joinings of Cartan Actions by Toral Automorphisms

We show that if $r\geq 3$ and $α$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $α$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.

math.DS↗

arXiv2010

Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent one-dimensional quantitative density result to abelian algebraic actions on higher dimensional tori. Up to finite index, the group actions that we study are conjugate to the action of $U_K$, the group of units of some non-CM number field $K$, on a compact quotient of $K\otimes_{\mathbb Q}\mathbb R$. In such a setting, we investigate how fast the orbit of a generic point can become dense in the torus. This effectivizes a special case of a theorem of Berend; and is deduced from a parallel measure-theoretical statement which effectivizes a special case of a result by Katok-Spatzier. In addition, we specify two numerical invariants of the group action that determine the quantitative behavior, which have number-theoretical significance.

math.DS↗