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

TL;DR

This paper generalizes Kronecker's approximation theorem by proving the existence of well-bounded lattice points outside algebraic sets that approximate given targets under linear forms, with explicit bounds depending on algebraic data.

Contribution

It introduces an effective version of Kronecker's theorem for algebraic lattices avoiding algebraic sets, with explicit bounds on the approximating vectors.

Findings

Existence of lattice points outside algebraic sets approximating targets.

Explicit bounds on the sup-norm of approximating vectors.

Generalization of Kronecker's approximation theorem to algebraic lattices.

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 $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ 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~$\mathbb R^t/\mathbb Z^t$.