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Register a new userCompact Open Spectral Sets In $mathbb{Q}_p$
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In this article, we prove that a compact open set in the field $mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $mathbb{Z}/p^n mathbb{Z}$ ($pge 2$ prime, $nge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $mathbb{Q}_p$, some of which are self-similar measures.
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Any bounded tile of the field $mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact.
Fugledes conjecture in $mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $mathbb{Q}_p$ is a spectral set if and only if it tiles $mathbb{Q}_p$ by translation.
Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.
We study the Chabauty compactification of two families of closed subgroups of $SL(n,mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, mathbb{Q}_p)$. The second family is the set of all $SL(n,mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $nleq 4$ (up to conjugacy). In contrast, for $ngeq 7$ we prove there are infinitely many $SL(n,mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $mathfrak{sl}(n, mathbb{Q}_p)$ of $SL(n,mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,mathbb{Q}_p)$-conjugates of $C$.
Subsets of the set of $g$-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, contained in the hull of a single point.
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