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

Abstract

Let Λ ⊂ R n \Lambda \subset \mathbb R^n be an algebraic lattice coming from a projective module over the ring of integers of a number field K K . Let Z ⊂ R n \mathcal Z \subset \mathbb R^n be the zero locus of a finite collection of polynomials such that Λ ⊈ Z \Lambda \nsubseteq \mathcal Z or a finite union of proper full-rank sublattices of Λ \Lambda . Let K 1 K_1 be the number field generated over K K by coordinates of vectors in Λ \Lambda , and let L 1 , … , L t L_1,\dots ,L_t be linear forms in n n variables with algebraic coefficients satisfying an appropriate linear independence condition over K 1 K_1 . For each ε > 0 \varepsilon > 0 and a ∈ R n \boldsymbol a \in \mathbb R^n , we prove the existence of a vector x ∈ Λ ∖ Z \boldsymbol x \in \Lambda \setminus \mathcal Z of explicitly bounded sup-norm such that ‖ L i ( x ) − a i ‖ > ε \begin{equation*} \| L_i(\boldsymbol x) - a_i \| > \varepsilon \end{equation*} for each 1 ≤ i ≤ t 1 \leq i \leq t , where ‖ ‖ \|\ \| stands for the distance to the nearest integer. The bound on sup-norm of x \boldsymbol x depends on ε \varepsilon , as well as on Λ \Lambda , K K , Z \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 ∖ Z \Lambda \setminus \mathcal Z under the linear forms L 1 , … , L t L_1,\dots ,L_t in the t t -torus R t / Z t \mathbb R^t/\mathbb Z^t .