The inverse problem for the lattice points

关于Minkowski球、域及其临界格的直接极限与逆向极限

On direct and inverse limits of Minkowski's balls, domains, and their critical lattices

本文构造Minkowski球、域及其临界格的直接与逆向系统，并计算相应的直接极限和逆向极限。

A note on the inverse problem for the lattice points

Let $K\subseteq\mathbb{R}^2$ be a compact set such that $K+\mathbb{Z}^2=\mathbb{R}^2$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap\mathbb{Z}^2$, is not contained on the coordinate axes, $\mathbb{Z}\times\{0\}\cup\mathbb{Z}\times\{0\}$. This result gives a negative answer to a question posed by P. Hegarty and M. Nathanson on relatively prime lattice points.

Estimates for the growth of inverse determinant sums of quasi-orthogonal and number field lattices

Inverse determinant sums appear naturally as a tool for analyzing performance of space-time codes in Rayleigh fading channels. This work will analyze the growth of inverse determinant sums of a family of quasi-orthogonal codes and will show that the growths are in logarithmic class. This is considerably lower than that of comparable number field codes.