English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\psi$ be a continuous decreasing function defined on all large positivereal numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet iffor every sufficiently large real number $t$ one can find $\boldsymbol{p} \in\mathbb{Z}^m$, $\boldsymbol{q} \in\mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying$\|A\boldsymbol{q}-\boldsymbol{p}\|^m< \psi({t})$ and$\|\boldsymbol{q}\|^n<{t}$. This property was introduced by Kleinbock andWadleigh in 2018, generalizing the property of $A$ being Dirichlet improvablewhich dates back to Davenport and Schmidt (1969). In the present paper, we givesufficient conditions on $\psi$ to ensure that the set of $\psi$-Dirichletmatrices has zero or full Lebesgue measure. Our proof is dynamical and relieson the effective equidistribution and doubly mixing of certain expandinghorospheres in the space of lattices. Another main ingredient of our proof isan asymptotic measure estimate for certain compact neighborhoods of thecritical locus (with respect to the supremum norm) in the space of lattices.Our method also works for the analogous weighted problem where the relevantsupremum norms are replaced by certain weighted quasi-norms.

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem