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  • Author or Editor: K. Fukuyama x
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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  

Takahashi [4] gave a concrete upper bound estimate of the law of the iterated logarithm for Σf(n k x). We extend this result and prove the best possibility of this bound.

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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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Abstract  

An upper bound estimate in the law of the iterated logarithm for Σf(n k ω) where nk+1∫nk≧ 1 + ck (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n k} is assumed. As an application, the law of the iterated logarithm is proved when {n k} is the arrangement in increasing order of the set B(τ)={1 i 1...qτ i τ|i1,...,iτN 0}, where τ≧ 2, N 0=NU{0}, and q 1,...,q τ are integers greater than 1 and relatively prime to each others.

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Kaufman and Philipp [6] and Dhompongsa [3] proved a uniform law of the iterated logarithm for ∑f(n k t). In this paper, we give a concrete upper bound for these results. Our bound is best possible in some cases.

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