Article - preprint
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 Λ, K, Z, heights of linear forms, and ε. 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 .
© 2016 Fukshansky
Fukshansky, Lenny, On an effective variation of Kronecker's approximation theorem, preprint.