We investigate the asymptotic distribution of perturbed geometric progression \(\{\theta^k x+ \gamma_k\}\) given by \(\theta\in (-\infty, -1)\cup (1, \infty)\) and \(\gamma_1, \gamma_2, \dots \in \mathbf {R}\). We prove that the discrepancy obeys the law of the iterated logarithm with limsup constant sensitively depending on \(\theta\) and \(\{\gamma_k \}\).

中文翻译：

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扰动几何级数差异的迭代对数定律

我们调查由（（\ theta \ in（-\ infty，-1）\ cup（1，\ infty）\）给出的摄动几何级数\（\ {\ theta ^ k x + \ gamma_k \} \）的渐近分布和\（\ gamma_1，\ gamma_2，\ dots \ in \ mathbf {R} \）。我们证明了差异遵循limsup常数，敏感地取决于\（\ theta \）和\（\ {\ gamma_k \} \）的迭代对数律。