Related papers: Almost Witt Vectors

Approximation in this paper is of vectors on the unit $d$-cube by the projection of integer lattice points onto the same cube. We define badly approximable vectors on a rational quadratic variety and show that sets of these vectors, which…

We describe rings over which every right module is almost injective. We give a description of rings over which every simple module is a almost projective.

The ring of Witt vectors associated to a ring R is a classical tool in algebra. We introduce a ring C(R) which is more easily constructed and which is isomorphic to the ring of Witt vectors W(R) for a perfect F_p-algebra R. It is obtained…

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of…

A subtraction-free definition of the big Witt vector construction was recently given by the first author. This allows one to define the big Witt vectors of any semiring. Here we give an explicit combinatorial description of the big Witt…

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings' almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and…

The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct…

We introduce quasi-Prufer extensions of rings in order to relativize the notion of quasi-Prufer domains and to take into account some contexts recently introduced in the literature. We also introduce almost-Prufer ring extensions.…

In this note, we investigate the Witt vectors in the min-plus algebra of extended non-negative real numbers, and consider categories enriched over them viewed as a monoidal category.

The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted A_inf, plays a pivotal role in p-adic Hodge theory; for instance, Bhatt, Morrow, and Scholze have recently reinterpreted and refined the crystalline…

Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…

Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of…

In this paper, we define vector bundles within the framework of almost mathematics (referred to as almost vector bundles) and establish the $v$-descent theorem together with a structure theorem for these bundles over perfectoid spaces. The…

For a profinite group $G$ we describe an abelian group $W_G(R; M)$ of $G$-typical Witt vectors with coefficients in an $R$-module $M$ (where $R$ is a commutative ring). This simultaneously generalises the ring $W_G(R)$ of Dress and…

In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the…

We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual…

We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic groups over global fields with respect to special sets of valuations. While…

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability.…

We give an abstract approach to the results of Adams and Nobel, [1]. It allows to exhibit a new property of VC classes. It should be stressed that the basic ideas of proofs can be found in [1].

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in…