664,932 research outputs found
Zero-one laws in simultaneous and multiplicative Diophantine approximation
Answering two questions of Beresnevich and Velani, we develop zero-one laws
in both simultaneous and multiplicative Diophantine approximation. Our proofs
rely on a Cassels-Gallagher type theorem as well as a higher-dimensional
analogue of the cross fibering principle of Beresnevich, Haynes and Velani
Some natural zero one laws for ordinals below ε0
We are going to prove that every ordinal α with ε_0 > α ≥ ω^ω satisfies a natural zero one law in the following sense. For α < ε_0 let Nα be the number of occurences of ω in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with ω ω  ≤ α < ε_0 and any sentence ϕ in the language of linear orders the asymptotic density of ϕ along α is an element of  {0,1}. We further show that for any such sentence ϕ the asymptotic density along ε_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω^ω
A note on zero-one laws in metrical Diophantine approximation
In this paper we discuss a general problem on metrical Diophantine
approximation associated with a system of linear forms. The main result is a
zero-one law that extends one-dimensional results of Cassels and Gallagher. The
paper contains a discussion on possible generalisations including a selection
of various open problems.Comment: 12 pages, Dedicated to Wolfgang Schmidt on the occasion of his 75th
birthda
Symmetry-breaking and zero-one laws
We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincaré invariance. In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions (corresponding respectively to breakings of reflection-invariance, translation-invariance, and Lorentz invariance). Along the way we provide a proof from first principles of the zero-one law on which our new arguments are based
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