Asymptotic estimate on the distance energy of lattices
DOI:
https://doi.org/10.55016/ojs/cdm.v20i2.76744Abstract
Since the well-known breakthrough of L. Guth and N. Katz on the Erdős distinct distances problem in the plane, it aroused mainstream interest by their method and the Elekes–Sharir framework. In short, they study the second moment in the framework. One may wonder if higher moments would be more efficient. In this paper, using number-theoretic methods, we show that any higher moment fails the expectation. We also show that the second moment gives an optimal estimate in higher dimensions. Moreover, we prove the mean second moment attains the maximum for the hexagonal lattice in $\mathbb{R}^2$, which is a parallel result on the distinct distances problem for lattices.
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