In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates only periodic tilings, while the second generates tilings for which any associated Delone set is non-equivalent to any lattice in space. As an extension of this result, we introduce arbitrarily many distinct substitution rules associated with a single matrix, with the property that Delone sets generated by distinct rules are non-equivalent. We then turn to the study of mixed substitution tilings, and present a mixed substitution system that generates representatives of continuously many distinct BD equivalence classes.

在研究准晶体的非周期性和数学模型时，自然会产生关于Delone集上等价关系的问题。这项工作致力于有界位移（BD）等价关系，尤其是有关不等价实例的结果。我们提出两个Delone集不等于BD的一般条件，并将我们的结果应用于与欧几里得空间平铺相关的Delone集。首先，我们考虑替换平铺，并展示与两个不同的替换规则相关的替换矩阵。第一个规则仅生成定期平铺，而第二个规则则生成任何关联的Delone集均不等于空间中的任何格的平铺。作为此结果的扩展，我们任意引入与单个矩阵相关的许多不同的替换规则，具有由不同规则生成的Delone设置的属性是不等价的。然后，我们转向混合替代拼贴的研究，并提出一种混合替代系统，该系统生成连续许多不同的BD等价类的代表。