Lower bounds for the discrepancy of some sequences. (English) Zbl 1108.11054

The authors study the sequence \(x_n=\alpha n+f(n)\bmod 1\), \(n=1,2,\dots \), where \(\alpha \) is an irrational number with bounded partial quotients and \(f(x)\) is a regularly varying function. Applying the saddle-point method to estimate the related trigonometric sum they obtain a lower bound for the extremal discrepancy \[ D_N(x_n)=\sup _{I\subset [0,1]}\left | \frac {\#\{n\leq N;x_n\in I\}}{N}-| I| \right | \] (\(I\) is a subinterval of \([0,1]\) and \(| I| \) is the length of \(I\)), which for \(f(x)=\beta (\log x)^s\), \((\beta >0, s\geq 1)\), implies \[ D_N\bigl (\alpha n+\beta (\log n)^s\bigr )\geq C_1\frac {(\log N)^{(s-1)/4}}{N^{3/4}} \] and for \(f(x)=\beta x^\sigma \), \((\beta >0, 0<\sigma <1)\) gives \[ D_N(\alpha n+\beta n^\sigma )\geq C_2N^{(\sigma -3)/4}, \] for all positive integers \(N\).
This completes previous results of the second author [J. Aust. Math. Soc., Ser. A 67, No. 1, 51–57 (1999; Zbl 0940.11029)] on upper bounds \(D_N(\alpha n+\beta \log n)\leq C_3\frac {\log N}{N^{2/3}}\) and \(D_N(\alpha n+\beta n^{\sigma })\leq C_4N^{-\sigma /2}\), \((0<\sigma <1/2)\). Basic properties of \(D_N\) can be found in L. Kuipers and H. Niederreiter [Uniform Distribution of Sequences, Wiley, New York (1974; Zbl 0281.10001); reprint edition, Dover Publications, New York (2006)].


11K38 Irregularities of distribution, discrepancy
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