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dc.contributor.authorFukshansky, Lenny
dc.contributor.authorGerman, Oleg
dc.contributor.authorMoshchevitin, Nikolay
dc.date.accessioned2017-10-25T07:57:32Z
dc.date.available2017-10-25T07:57:32Z
dc.date.issued2017-10-19
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1316
dc.descriptionResearch in Pairs 2017en_US
dc.description.abstractLet $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$. 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.en_US
dc.language.isoen_USen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2017,28
dc.subjectKronecker's Theoremen_US
dc.subjectDiophantine Approximationen_US
dc.subjectHeightsen_US
dc.subjectPolynomialsen_US
dc.subjectLatticesen_US
dc.titleOn an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Setsen_US
dc.typePreprinten_US
dc.rights.licenseDieses 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.de
dc.rights.licenseThis 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.en
dc.identifier.doi10.14760/OWP-2017-28
local.scientificprogramResearch in Pairs 2017en_US
local.series.idOWP-2017-28
local.subject.msc11
dc.identifier.urnurn:nbn:de:101:1-2017111411281
dc.identifier.ppn1658648250


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