ON AN EFFECTIVE VARIATION OF KRONECKER'S APPROXIMATION THEOREM AVOIDING ALGEBRAIC SETS

Let Lambda subset of R-n be an algebraic lattice coming from a projective module over the ring of integers of a number field K. Let Z subset of R-n be the zero locus of a finite collection of polynomials such that Lambda not subset of 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,..., L-t be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K-1. For each epsilon > 0 and a is an element of R-n, we prove the existence of a vector x is an element of Lambda \ Z of explicitly bounded sup-norm such that parallel to Li(x) - a(i)parallel to < epsilon for each 1 <= i <= t, where parallel to parallel to stands for the distance to the nearest integer. The bound on sup-norm of x depends on epsilon, as well as on Lambda, 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 Lambda \ Z under the linear forms L-1,..., L-t in the t-torus R-t/Z(t).