dc.contributor.author Fukshansky, Lenny dc.contributor.author German, Oleg dc.contributor.author Moshchevitin, Nikolay dc.date.accessioned 2017-10-25T07:57:32Z dc.date.available 2017-10-25T07:57:32Z dc.date.issued 2017-10-19 dc.identifier.uri http://publications.mfo.de/handle/mfo/1316 dc.description Research in Pairs 2017 en_US dc.description.abstract Let $\Lambda \subset \mathbb R^n$ 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 en_US $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ 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. dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2017,28 dc.subject Kronecker's Theorem en_US dc.subject Diophantine Approximation en_US dc.subject Heights en_US dc.subject Polynomials en_US dc.subject Lattices en_US dc.title On an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Sets en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2017-28 local.scientificprogram Research in Pairs 2017 en_US local.series.id OWP-2017-28 local.subject.msc 11
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