Lower bounds for diophantine approximations

Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

Given n ∈ N and τ > 1 n , let Sn(τ) denote the classical set of τ approximable points in R, which consists of x ∈ R that lie within distance q from the lattice 1 q Z for infinitely many q ∈ N. In pioneering work, Kleinbock & Margulis showed that for any non-degenerate submanifold M of R and any τ > 1 n almost all points on M are not τ -approximable. Numerous subsequent papers have been geared t...

Some Lower Bounds for Estrada Index

Diophantine Approximations, Diophantine Equations, Transcendence and Applications

This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...

Errors Bounds for Finite Approximations of Coherent Lower Previsions

Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabil...

Lower Bounds for Approximations by Low Degree Polynomials Over

Number 98-36Graph Ramsey Theory and the Polynomial HierarchybyMarcus SchaeferDepartment of Computer ScienceUniversity of Chicago1100 East 58th StreetChicago, Illinois 60637, [email protected] the Ramsey Theory of graphs F ! (G;H) means that for everycoloring of F with colors red and blue there is either a red subgraph Gor a blue su...