Elements in a less than a value in a subarray
This is essentially the 2D orthogonal range counting problem from computational geometry, which has attracted a lot of attention over the years. Fractional cascading definitely could be applied to reduce the query time to logarithmic while preserving the other big-O bounds. In a nutshell, the idea is to make auxiliary data structures that relate the positions of elements in each parent array to positions in the child array, so that, given the results of a search in the parent, searching the child for the same key is constant-time.