# Andreas Wieser

I am a mathematician. My work is mostly in homogeneous dynamics and its applications in number theory.

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# Research

Here is a list of my publications from new to old.

1. O.Solan and A.Wieser. Birkhoff generic points on curves in horospheres . 2023, arXiv:2301.10671.
Abstract Let $\{a_t:t∈\R\}<SL_d(\R)$ be a diagonalizable subgroup whose expanding horospherical subgroup $U<SL_d(\R)$ is abelian. By the Birkhoff ergodic theorem, for any $x\in \SL_d(\R)/SL_d(\Z)$ and for almost every point $u\in U$ the point $ux$ is Birkhoff generic for $a_t$ when $t\to \infty$. We prove that the same is true when $U$ is replaced by any non-degenerate analytic curve in $U$. This Birkhoff genericity result has various applications in Diophantine approximation. For instance, we obtain density estimates for Dirichlet improvability along typical points on a curve in Euclidean space. Other applications address approximations by algebraic numbers and best approximations (in the sense of Lagarias).
2. A. Wieser and P.Yang. A uniform Linnik basic lemma and entropy bounds . 2022, arXiv:2201.05380.
Abstract We prove a version of Linnik's basic lemma uniformly over the base field using theta-series and geometric invariant theory in the spirit of Khayutin's approach (Duke Math. J., 168(12), 2019). As an application, we establish entropy bounds for limits of invariant measures on homogeneous toral sets in GL(4) of biquadratic, cyclic, or dihedral type.
3. M. Aka, A. Musso, and A. Wieser. Equidistribution of rational subspaces and their shapes . 2021, arXiv:2103.05163.
Abstract To any $k$-dimensional subspace of $\mathbb{Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm{Gr}_{n,k}(\mathbb{R})$ and two shapes of lattices of rank $k$ and $n-k$ respectively. These lattices originate by intersecting the $k$-dimensional subspace with the lattice $\mathbb{Z}^n$. Using unipotent dynamics we prove simultaneous equidistribution of all of these objects under a congruence conditions when $(k,n)\neq (2,4)$.
4. M. Aka, M. Luethi, Ph. Michel and A. Wieser. Simultaneous supersingular reductions of CM elliptic curves . J. Reine Angew. Math. 786 (2022), 1-43.
Abstract We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show - under additional congruence assumptions on the CM order - that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.
Here is a talk by my coauthor Manuel Luethi on the topic.
5. M. Aka, M. Einsiedler and A. Wieser. Planes in four space and four associated CM points. Duke Math. J. 171 (2022), no. 7, 1469-1529.
Abstract To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian $\operatorname{Gr}(2,4)$ and four lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement and the second two essentially arise from the accidental local isomorphism between $\operatorname{SO}(4)$ and $\operatorname{SO}(3)\times \operatorname{SO}(3)$. As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings we prove simultaneous equidistribution of all of these objects under two splitting conditions.
Here are the slides for this talk I gave at the conference 'Smooth and homogeneous dynamics' at ICTS in Bangalore.
6. A. Wieser. Linnik's problems and maximal entropy methods. Monatsh. Math. 190 (2019), 153-208.
Abstract We use maximal entropy methods to examine the distribution properties of primitive integer points on spheres and of CM points on the modular surface. The proofs we give are a modern and dynamical interpretation of Linnik's original ideas and follow techniques presented by Einsiedler, Lindenstrauss, Michel and Venkatesh in 2011.
Note: the first version on the ArXiv treats a simpler special case.