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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.identifier.doi10.14760/OWP-2017-28
local.scientificprogramResearch in Pairs 2017en_US
local.series.idOWP-2017-28
local.subject.msc11


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