Some problems of unlikely intersections

Fixing X in the multiplicative algebraic group G^n_m(\Q), we are interested in the intersection of X with the union of Y where Y runs through the algebraic subgroups of G^n_m restricted only by dimension. We shall prove that, if X is an irreducible subvariety in G^n_m and the considered algebraic subgroups have codimension at least dim X, then this intersection has a bounded height by removing the anomalous sub varieties of X. Furthermore, we recover the finiteness by considering the algebraic subgroups of codimension at least 1 + dim X.