On an effective variation of Kronecker’s approximation theorem avoiding algebraic sets
dc.bibliographicCitation.seriesTitle | Oberwolfach Preprints (OWP) | eng |
dc.bibliographicCitation.volume | 2017-28 | |
dc.contributor.author | Fukshansky, Lenny | |
dc.contributor.author | German, Oleg | |
dc.contributor.author | Moshchevitin, Nikolay | |
dc.date.accessioned | 2017-11-24T21:32:57Z | |
dc.date.available | 2019-06-28T08:08:08Z | |
dc.date.issued | 2017 | |
dc.description.abstract | 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. | eng |
dc.description.version | publishedVersion | eng |
dc.format | application/pdf | |
dc.identifier.issn | 1864-7596 | |
dc.identifier.uri | https://doi.org/10.34657/2814 | |
dc.identifier.uri | https://oa.tib.eu/renate/handle/123456789/2558 | |
dc.language.iso | eng | eng |
dc.publisher | Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach | eng |
dc.relation.doi | https://doi.org/10.14760/OWP-2017-28 | |
dc.rights.license | This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. | eng |
dc.rights.license | Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. | ger |
dc.subject.ddc | 510 | eng |
dc.subject.other | Kronecker's theorem | eng |
dc.subject.other | Diophantine approximation | eng |
dc.subject.other | heights | eng |
dc.subject.other | polynomials | eng |
dc.subject.other | lattices | eng |
dc.title | On an effective variation of Kronecker’s approximation theorem avoiding algebraic sets | eng |
dc.type | Report | eng |
dc.type | Text | eng |
tib.accessRights | openAccess | eng |
wgl.contributor | MFO | eng |
wgl.subject | Mathematik | eng |
wgl.type | Report / Forschungsbericht / Arbeitspapier | eng |
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