Abstract Let α , β ∈ ( 0 , 1 ) such that at least one of them is irrational. We take a random walk on the real line such that the choice of α and β has equal probability 1/2. We prove that almost surely the αβ-orbit is uniformly distributed module one, and the exponential sums along its orbit have the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property of uniform distribution modulo one, is large in the terms of topology and Hausdorff dimension.

中文翻译：

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关于αβ轨道的均匀分布

摘要 令 α , β ∈ ( 0 , 1 ) 使得其中至少一个是无理数。我们在实线上随机游走，使得 α 和 β 的选择具有相等的概率 1/2。我们几乎可以肯定地证明 αβ 轨道是均匀分布的模 1，并且沿其轨道的指数和具有平方根相消。我们还表明，概率空间中的异常集不具有均匀分布模 1 的性质，在拓扑和 Hausdorff 维数方面很大。