Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term “Fourier sums” should be understood as “partial Fourier integrals”. A certain weighted version of the result is also provided. This version leads to a new Men\('\)shov-type correction theorem.

中文翻译：

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具有均匀界傅里叶和和频谱中大间隙的指标函数

标题中提到的指标函数是在任意非离散局部紧凑的Abelian有限维组上构建的。此外，它们可以通过预先确定的任何指示器功能的微小扰动来获得。对于非紧群，术语“傅立叶和”应理解为“偏傅立叶积分”。还提供了结果的某个加权版本。这个版本导致了一个新的Men \（'\） shov型校正定理。