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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}. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]