arXiv subjects

René Pfitscher

Publications and source records attributed to René Pfitscher.

3 recordsLinked to original sources

arXiv2026

Zero-one laws for uniform approximation via Gaussian and Eisenstein integers

We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing it to a shrinking target problem on certain homogeneous spaces of $\mathrm{SL}_2(\mathbb{C})$. The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.

math.DS↗

arXiv2026

Counting rational approximations on rank one flag varieties

On a generalized flag variety of rank one, we count rational approximations to a real point chosen randomly according to the Riemannian volume. In particular, our results apply to Grassmann varieties and quadric hypersurfaces. The proof uses exponential mixing in the space of lattices and tools from geometry of numbers.

math.NT↗

arXiv2025

Integrability of Siegel transforms and an application

We establish sharp algebraic criteria for the $L^{p}$-integrability, for $p = 1, 2, \infty$, of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic $\mathbb{Q}$-groups, extending Siegel's analytic work in the geometry of numbers. As an application, we derive an effective asymptotic formula for the number of rational approximations of bounded height to almost every real point on a rank-one flag variety at the Diophantine exponent. The argument combines the integrability criterion with effective equidistribution estimates for translated orbits of maximal compact subgroups, a result of independent interest.

math.NT↗