  This paper develops the theory of pushdown dimension and explores its
relationship with finite-state dimension. Pushdown dimension is trivially
bounded above by finite-state dimension for all sequences, since a pushdown
gambler can simulate any finite-state gambler. We show that for every rational
0 < d < 1, there exists a sequence with finite-state dimension d whose pushdown
dimension is at most d/2. This establishes a quantitative analogue of the
well-known fact that pushdown automata decide strictly more languages than
finite automata.
