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arXiv · 1604.05989

Finding well approximating lattices for a finite set of points

A. Hajdu·L. Hajdu·R. Tijdeman
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Abstract

In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $\v{o},\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $\v{a}-\v{o}$ is close to $\Lambda=\v{d_1}\mathbb{Z}+\dots+\v{d_n}\mathbb{Z}$ for every $\v{a}\in A$. First we deal with the one-dimensional case, where we show that in a sense the results are almost the best possible. These results easily extend to the multi-dimensional case where the directions of the axes are given, too. Thereafter we treat the general multi-dimensional case. Our method relies on the LLL algorithm. Finally we apply the least squares algorithm to optimize the results. We give several examples to illustrate our approach.

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A. Hajdu, L. Hajdu, R. Tijdeman. 2016-04-20. Finding well approximating lattices for a finite set of points. https://arxiv.org/abs/1604.05989

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