On an effective variation of Kronecker's approximation theorem avoiding algebraic sets

Abstract

Let $Λ\subset \mathbb R^n$ 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 $Λ\nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $Λ$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $Λ$, 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 $\boldsymbol x \in Λ\setminus \mathcal Z$ 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 $Λ$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $Λ\setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$.