  We study the set of eventually always hitting points for symbolic dynamics
with specification. The measure and Hausdorff dimension of such sets are
obtained. Moreover, we establish the stronger metric results by introducing a
new quantity L_{N}({\omega}) which describes the maximal length of string of
zreos of the prefix among the first N iterations of {\omega} in symbolic space.
The Hausdorff dimensions of the level sets for this quantity are also
completely determined.
