The law of the iterated logarithm for the discrepancies of a permutation of {n k x}

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 }.

The central limit theorem for complex Riesz–Raikov sums

“…To prove (6), we use the following result. The one-dimensional version is proved in [5][6][7]. The following version can be proved in the same way as [11].…”

Metric discrepancy results for alternating geometric progressions

“…We have already proved in [15] that σ |θ|;a ,a is continuous with respect to (a , a) ∈ T 2 . Thus the equality (7) follows from (17).…”

“…Before closing introduction, we mention results relating to permutations of sequences. In [17] it was found that the limsups are not invariant under permutations of sequences, and this phenomenon is studied extensively by Aistleitner-Berkes-Tichy [5][6][7][8][9].…”

On permutations of Hardy-Littlewood-Pólya sequences

“…For example, for n k = q k , q ≥ 2, the limsup Σ q in (1. Even more surprisingly, Fukuyama [7] showed that the limsup Σ in (1.1) is not permutation-invariant: changing the order of the (n k ) generally changes the value of Σ. This is quite unexpected, since {n k x} are identically distributed in the sense of probability theory and the asymptotic properties of i.i.d.…”