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

Abstract

Let Λ⊂Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z⊂Rn be the zero locus of a finite collection of polynomials such that Λ⊈Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1,…,Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε>0 and a∈Rn, we prove the existence of a vector x∈Λ∖Z of explicitly bounded sup-norm such that ∥Li(x)−ai∥<ε for each 1≤i≤t, where ∥ ∥ stands for the distance to the nearest integer. The bound on sup-norm of x depends on ε, as well as on Λ, K, 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 under the linear forms L1,…,Lt in the t-torus~Rt/Zt. 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.