Let u be a solution to an elliptic equation div(Aru) = 0 with Lipschitz coefficients in Rn. Assume juj is bounded by 1 in the ball B = fjxj ≥ 1g. We show that if juj < " on a set E ⊂ 21 B with positive n-dimensional Hausdorf measure, then 1 juj ≤ C "γ on 2 B; where C > 0; γ 2 (0; 1) do not depend on u and depend only on A and the measure of E. We specify the dependence on the measure of E in the form of the Remez type inequality. Similar estimate holds for sets E with Hausdorff dimension bigger than n = 1. For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than n = 1 = c, where c > 0 is a small numerical constant depending on the dimension only.