Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue–Morse measure.

中文翻译：

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关于与加倍映射相关的奇异连续测度族

在这里，我们研究了一些可以由 1-周期函数的无限 Riesz 积表示的度量，并且与倍增映射相关。我们表明，这些测度相对于 Lebesgue 测度而言是纯奇异连续的，并且它们的分布函数满足原点附近的超多项式渐近性，从而提供了一系列奇异测度的极值示例，包括 Thue-Morse 测度。