Recent zbMATH articles in MSC 37A17https://zbmath.org/atom/cc/37A172024-03-13T18:33:02.981707ZUnknown authorWerkzeugGroup actions and harmonic analysis in number theory. Abstracts from the workshop held May 7--12, 2023https://zbmath.org/1528.110012024-03-13T18:33:02.981707ZSummary: This workshop focuses on new problems and new methods at the interface of harmonic analysis (taken in a very broad sense) and ergodic theory, with applications focused on number theory. Special emphasis is put on equidistribution problems on arithmetic symmetric spaces, effective methods in homogeneous dynamics, periods of automorphic forms, families of \(L\)-functions over number fields and function fields, and applications of Fourier uniqueness.Singular vectors on manifolds over totally real number fieldshttps://zbmath.org/1528.110562024-03-13T18:33:02.981707Z"Datta, Shreyasi"https://zbmath.org/authors/?q=ai:datta.shreyasi"Radhika, M. M."https://zbmath.org/authors/?q=ai:radhika.m-mDirichlet's approximation theorem for a single linear form states that for any \(x \in\mathbb{R}^m\) and any large integer \(T\), there is a \(p\in\mathbb{Z}\) and a \(q \in \mathbb{Z}^m\setminus \{0\}\) with \(\vert q \cdot x - p \vert < 1/T^m\) and \(\Vert q \Vert \le T\). Here, \(\Vert \cdot \Vert\) denotes the supremum norm. A vector is said to be singular if for any \(c > 0\), for any sufficiently large \(T\), one can improve these inequalities to \(\vert q \cdot x - p \vert < c/T^m\) and \(\Vert q \Vert \le T\).
In the present paper, these notions are extended to approximation by numbers from a fixed totally number field and several results results from the classical theory are extended to this setting. Let \(K\) be a totally real number field of degree \(d\), let \(\mathcal{O}_K\) be its ring of integers and let \(S\) be the normalised inequivalent valuations of \(K\). For each \(\sigma \in S\), let \(K_\sigma \cong \mathbb{R}\) denote the completion of \(K\) at \(\sigma\) and let \(K_S = \prod_{\sigma \in S} K_\sigma\). Identify \(K\) with its diagonal embedding into \(K_S\), and for \(m >1\), identify \(K^m\) with the coordinate wise diagonal embedding of \(K\) into \(K_S^m\). In this setting, a vector \(x \in K_S^m\) is singular if for any \(c>0\), for \(Q>0\) sufficiently large, there is a \(q_0 \in \mathcal{O}_K\) and \(q \in \mathcal{O}_K^m \setminus \{0\}\), such that \(\Vert q \cdot x - q_0 \Vert < c/Q^m\) and \(\Vert q \Vert \le Q\).
The authors prove that this notion is indeed the correct extension of the notion of singular vectors by proving a version of Dirichlet's theorem. They subsequently prove that the set of singular measures is a null set with respect to a large class of measures -- the so-called friendly measures -- which include the natural measures on a large family of non-degenerate manifolds as well as many fractal measures. They also prove, that if \(m=1\) the set of singular elements in \(K_S\) coincides with the field \(K\), while for \(m > 1\), singular vectors exist in abundance. The results are extensions of known results for the case \(K= \mathbb{Q}\). They are proved using homogeneous dynamics via establishing an analogue of Mahler's compactness criterion and Dani's correspondence in the present setup.
Reviewer: Simon Kristensen (Aarhus)Quantitative disjointness of nilflows from horospherical flowshttps://zbmath.org/1528.370052024-03-13T18:33:02.981707Z"Katz, Asaf"https://zbmath.org/authors/?q=ai:katz.asafSummary: We prove a quantitative variant of a disjointness theorem of nilflows from horospherical flows following a technique of \textit{A. Venkatesh} [Ann. Math. (2) 172, No. 2, 989--1094 (2010; Zbl 1214.11051)], combined with the structural theorems for nilflows by \textit{B. Green} et al. [Ann. Math. (2) 176, No. 2, 1231--1372 (2012; Zbl 1282.11007)].