A metric discrepancy result with given speed

Abstract

Abstract. It is known that the discrepancy D N {kx} of the se-a.e. for some 0 < Σ θ < ∞ and N ≥ N 0 if ε > 0, but not for ε < 0. In this paper we prove, extending results of Aistleitner-Larcher [6], that for any sufficiently smooth intermediate speed Ψ(N ) between (log N )(log log N ) 1+ε and (N log log N ) 1/2 and for any Σ > 0, there exists a sequence {n k } of positive integers such that N D N {n k x} ≤ (Σ + ε)Ψ(N ) eventually holds a.e. for ε > 0, but not for ε < 0. We also consider a similar problem on the growth of trigonometric sums.

Additive Energy and Irregularities of Distribution

“…Then there exists a strictly increasing sequence (a n ) n≥1 of positive integers such that for the discrepancy of ({a n α}) n≥1 for almost all α we have ND * N = O (N γ ) and ND * N = Ω (N γ−ε ) for all ε > 0. An even more precise result has been recently obtained by Berkes, Fukuyama and Nishimura [7], by using a randomization technique.…”

“…Further examples of sequences (a n ) n≥1 with highest possible additive energy and the essentially maximal metric order of ND * N of ({a n α}) n≥1 can be deduced from the results in [1,2,7], where it may be necessary to modify the sequences constructed there by inserting long stretches of arithmetic progressions in order to maximize the additive energy. However, the examples given in these papers are randomly generated sequences and no explicit constructions are known, whereas the example in Theorem 4 above is fully explicit.…”