By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.

中文翻译：

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超越比蒂序列：互补格

通过将方形格子作为比蒂序列的二维模拟，我们有动力定义和探索互补格子的概念。特别是，我们提出了一个连续的单参数互补晶格族。这个主要结果产生了几个互补序列的新例子，以及比蒂序列基本性质的几何证明。