  We prove explicit doubling inequalities and obtain uniform upper bounds
(under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions
for a family of linear elliptic equations with rapidly oscillating periodic
coefficients. The doubling inequalities, explicitly depending on the doubling
index, are proved at different scales by a combination of convergence rates, a
three-ball inequality from certain "analyticity", and a monotonicity formula of
a frequency function. The upper bounds of nodal sets are shown by using the
doubling inequalities, approximations by harmonic functions and an iteration
argument.
