  Since the well-known breakthrough of L. Guth and N. Katz on the Erdos
distinct distances problem in the plane, mainstream of interest is aroused by
their method and the Elekes-Sharir framework. In short words, they study the
second moment in the framework. One may wonder if higher moments would be more
efficient. In this paper, we show that any higher moment fails the expectation.
In addition, we show that the second moment gives optimal estimate in higher
dimensions.
