  In the orthogonal range reporting problem, we are to preprocess a set of $n$
points with integer coordinates on a $U \times U$ grid. The goal is to support
reporting all $k$ points inside an axis-aligned query rectangle. This is one of
the most fundamental data structure problems in databases and computational
geometry. Despite the importance of the problem its complexity remains
unresolved in the word-RAM. On the upper bound side, three best tradeoffs
exists: (1.) Query time $O(\lg \lg n + k)$ with $O(nlg^{\varepsilon}n)$ words
of space for any constant $\varepsilon>0$. (2.) Query time $O((1 + k) \lg \lg
n)$ with $O(n \lg \lg n)$ words of space. (3.) Query time
$O((1+k)\lg^{\varepsilon} n)$ with optimal $O(n)$ words of space. However, the
only known query time lower bound is $\Omega(\log \log n +k)$, even for linear
space data structures.
  All three current best upper bound tradeoffs are derived by reducing range
reporting to a ball-inheritance problem. Ball-inheritance is a problem that
essentially encapsulates all previous attempts at solving range reporting in
the word-RAM. In this paper we make progress towards closing the gap between
the upper and lower bounds for range reporting by proving cell probe lower
bounds for ball-inheritance. Our lower bounds are tight for a large range of
parameters, excluding any further progress for range reporting using the
ball-inheritance reduction.
