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Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from aprojective 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 ofpolynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union ofproper full-rank sublattices of $\Lambda$. Let $K_1$ be the number fieldgenerated over $K$ by coordinates of vectors in $\Lambda$, and let$L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficientssatisfying an appropriate linear independence condition over $K_1$. For each$\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existenceof a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitlybounded sup-norm such that $$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$ foreach $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearestinteger. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, aswell as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. Thispresents a generalization of Kronecker's approximation theorem, establishing aneffective result on density of the image of $\Lambda \setminus \mathcal Z$under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbbZ^t$.

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