相关论文: Vaaler's theorem in number fields

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine $m$-tuples in finite fields, which is due to A. Dujella and M.Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).

We prove a Torelli-like theorem for higher-dimensional function fields, from the point of view of "almost-abelian" anabelian geometry.

A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give…

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a…

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a…

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly…

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a…

We present the proof of the equivalence theorem in quantum field theory which is based on a formulation of this problem in the field-antifield formalism. As an example, we consider a model in which a different choices of natural finite…

Assuming the Generalized Riemann Hypothesis we obtain uniform, effective number-field analogues of Mertens' theorems.

We provide an improved version of the Darling-Erd\"os theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance…

We extend the Brauer-Siegel theorem to new families of number fields, both in the classical setting of asymptotically bad families and in the more general framework due to Tsfasman and Vl\u{a}du\c{t} of asymptotically exact families. We…

In this paper we present a new approach to prove effective results in Diophantine approximation. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with…

We prove a Szemer\'edi-Trotter type theorem and a sum-product estimate in the setting of finite quasifields. These estimates generalize results of the fourth author, of Garaev, and of Vu. We generalize results of Gyarmati and S\'ark\"ozy on…

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fr\'echet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.

In 1986, Kato and Kuzumaki stated several conjectures in order to give a diophantine characterization of cohomological dimension of fields. In this article, we first prove a local-global principle in this context for number fields. This…

Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an…

We extend the decomposition theorem for numerically $K$-trivial varieties with log terminal singularities to the K\"ahler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus…

The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations…

A gap in the mathematical logic in derivations in quantum field theory arises as consequence of variation before quantization. To close this gap the present paper introduces a mathematically rigorous variational calculus for operator…