  The paper is mainly concerned with an approximate three-ball inequality for
solutions in elliptic periodic homogenization. We consider a family of second
order operators $\mathcal{L}_\epsilon$ in divergence form with rapidly
oscillating and periodic coefficients. It is the first time such an approximate
three-ball inequality for homogenization theory is obtained. It implies an
approximate quantitative propagation of smallness. The proof relies on a
representation of the solution by the Poisson kernel and the Lagrange
interpolation technique. Another full propagation of smallness result is also
shown.
