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Hexagonal Lattice Points on Circles

Oscar Marmon
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Abstract

We study the hexagonal lattice $\mathbb{Z}[ω]$, where $ω^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on average, and suggest the possibility of constructing a consistent discrete velocity model (DVM) for the Boltzmann equation, using a hexagonal lattice. Equidistribution on average is expressed in terms of cancellation in exponential sums. We introduce Hecke L-functions and investigate their analytic properties in order to derive estimates on sums of Hecke characters. Using a version of the Halberstam-Richert inequality, these estimates then yield the desired results for the exponential sums. As a further measure of equidistribution, we give a bound for the discrepancy.

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Oscar Marmon. 2005-08-11. Hexagonal Lattice Points on Circles. https://arxiv.org/abs/math/0508201

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