  The rank problem in succinct data structures asks to preprocess an array
A[1..n] of bits into a data structure using as close to n bits as possible, and
answer queries of the form rank(k) = Sum_{i=1}^k A[i]. The problem has been
intensely studied, and features as a subroutine in a majority of succinct data
structures.
  We show that in the cell probe model with w-bit cells, if rank takes t time,
the space of the data structure must be at least n + n/w^{O(t)} bits. This
redundancy/query trade-off is essentially optimal, matching our upper bound
from [FOCS'08].
