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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References
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Fukuyama, K. The law of the iterated logarithm for the discrepancies of a permutation of {n k x}. Acta Math Hung 123, 121–125 (2009). https://doi.org/10.1007/s10474-008-8067-9
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DOI: https://doi.org/10.1007/s10474-008-8067-9