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- W2536364784 abstract "It is known that the discrepancy $${D_N{kx}}$$ of the sequence $${{kx}}$$ satisfies $${ND_N{kx} = O((log N){(log log N)}^{1+varepsilon})}$$ a.e. for all $${varepsilon > 0}$$ , but not for $${varepsilon=0}$$ . For $${n_k=theta^k}$$ , $${theta > 1}$$ we have $${ND_N{n_kx} leqq (Sigma_theta +varepsilon){(2Nlog log N)}^{1/2}}$$ a.e. for some $${0 < Sigma_theta < infty}$$ and $${Ngeqq N_0}$$ if $${varepsilon > 0}$$ , but not for $${varepsilon < 0}$$ . In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed $${Psi(N)}$$ between $${(log N){(log log N)}^{1+varepsilon}}$$ and $${{(Nlog log N)}^{1/2}}$$ and for any $${Sigma > 0}$$ , there exists a sequence $${{n_k}}$$ of positive integers such that $${ND_N{n_kx} leqq (Sigma+varepsilon)Psi(N)}$$ eventually holds a.e. for $${varepsilon > 0}$$ , but not for $${varepsilon < 0}$$ . We also consider a similar problem on the growth of trigonometric sums." @default.
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- W2536364784 date "2016-10-24" @default.
- W2536364784 modified "2023-10-17" @default.
- W2536364784 title "A metric discrepancy result with given speed" @default.
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- W2536364784 doi "https://doi.org/10.1007/s10474-016-0658-2" @default.
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