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Abstract  

For any unbounded sequence {n k} of positive real numbers, there exists a permutation {n σ(k)} such that the discrepancies of {n σ(k) x} obey the law of the iterated logarithm exactly in the same way as the uniform i.i.d. sequence {U k}.

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Abstract  

It is proved that two types of discrepancies of the sequence {θ n x} obey the law of the iterated logarithm with the same constant. The appearing constants are calculated explicitly for most of θ > 1.

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