Browsing by Author "Moshchevitin, Nikolay"

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    An Optimal Bound for the Ratio Between Ordinary and Uniform Exponents of Diophantine Approximation
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Marnat, Antoine; Moshchevitin, Nikolay
    We provide a lower bound for the ratio between the ordinary and uniform exponents of both simultaneous Diophantine approximation to n real numbers and Diophantine approximation for one linear form in n variables. This question was first considered in the 50's by V. Jarník who solved the problem for two real numbers and established certain bounds in higher dimension. Recently different authors reconsidered the question, solving the problem in dimension three with different methods. Considering a new concept of parametric geometry of numbers, W. M. Schmidt and L. Summerer conjectured that the optimal lower bound is reached at regular systems. It follows from a remarkable result of D. Roy that this lower bound is then optimal. In the present paper we give a proof of this conjecture by W. M. Schmidt and L. Summerer.
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    On an effective variation of Kronecker’s approximation theorem avoiding algebraic sets
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Fukshansky, Lenny; German, Oleg; Moshchevitin, Nikolay
    Let Λ⊂Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z⊂Rn be the zero locus of a finite collection of polynomials such that Λ⊈Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1,…,Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε>0 and a∈Rn, we prove the existence of a vector x∈Λ∖Z of explicitly bounded sup-norm such that ∥Li(x)−ai∥<ε for each 1≤i≤t, where ∥ ∥ stands for the distance to the nearest integer. The bound on sup-norm of x depends on ε, as well as on Λ, K, Z and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of Λ∖Z under the linear forms L1,…,Lt in the t-torus~Rt/Zt. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.