We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, what makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps.

中文翻译：

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采样几乎周期性和相关的函数

我们考虑在实数区间上定义的某些有限圆值函数集，并估计区间必须有多大才能使这些函数的值以近似方式均匀分布。这用于建立一些一般条件，在这些条件下，由 Katznelson 为整数引入的随机构造产生在 Bohr 群中稠密的集合。我们以这种方式获得非常稀疏的实数（和整数）集，两个不同的几乎周期函数在这两个函数上不能达成一致，这使得它们适合用于这些函数的采样定理。这些集合可以稀疏到具有零渐近密度或 t 集，即与它们在有界集合中的任何平移相交的集合。