## 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)].

### MSC:

 11K38 Irregularities of distribution, discrepancy

### Citations:

Zbl 0940.11029; Zbl 0281.10001
Full Text:

### References:

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