We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form x 2+ Dy 2= z 2 where D\u3e0 is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.
展开▼
机译：我们研究平面上积分良好的圆形晶格的分布，并通过x 2+ Dy 2 = z 2形式的Pell型Diophantine方程族的解参数化其相似类集，其中D \ u3e0是无平方的。我们将此参数化应用于具有固定行列式的此类晶格集上的最大最小范数和最高信噪比的研究，同时还估计了每个行列式值的这些集的基数（包括旋转和反射）。这项研究扩展了第一作者在整数和六边形晶格的特定情况下的先前工作，并且受到积分良好的圆形晶格对于离散优化问题的重要性的启发。我们简要讨论了我们的结果在平面晶格发射机网络中的应用。
展开▼
