Let f:X→S be a quasi-projective family of varieties defined over Q¯¯¯¯⊂C. We show that the points of S(Q¯¯¯¯) that are Hodge generic for the variation of Hodge structures associated to f are analytically dense in S(C). In fact, in the spirit of the Grothendieck period conjecture and under a large monodromy assumption, we prove the density of the points of S(Q¯¯¯¯) where the periods of the fibre do not satisfy extra relations ``up to degree δ''. As a by-product, we also establish new instances of the Mumford-Tate conjecture, beyond the realm of abelian motives. When the base S is a curve, we provide quantitative estimates for points satisfying these properties. The main technical contribution is a new result on relations satisfied by solutions of G-operators, which relies on height estimates due to Bombieri and Andr\'e. Joint work with G. Binyamini and D. Urbanik.