## Some remarks on the discrepancy of the sequence $$\bigl (\alpha \sqrt {n}\bigr)$$.(English)Zbl 1024.11053

An explicit formula concerning $$\liminf _{N\to \infty }N^{-1/2}D^+_N(\alpha)$$ of the one-sided discrepancy function $$D^+_N(\alpha)=\sup _{0\leq x<1} (\sum _{n=1}^Nc_{[0,x)}(\{\alpha \sqrt {n}\})-Nx)$$ is established for $$\alpha =\sqrt {p/q}$$, where $$p,q$$ are positive integers, $$c_{[0,x)}(t)$$ denotes the characteristic function of the interval $$[0,x)$$ and $$\{x\}$$ the fractional part of $$x$$. The idea of the proof is based on a previous work of the author and J. Schoißengeier [J. Lond. Math. Soc. (2) 57, 529-544 (1998; Zbl 0938.11041)] on $$\limsup _{N\to \infty }N^{-1/2}D^+_N(\alpha)$$. The result is too complicated to be reproduced here, but the author obtains exact values for special rationals $$p/q$$, for example if $$q=3$$ and $$p\equiv 2\pmod 3$$, then $$\liminf _{N\to \infty }N^{-1/2}D^+_N(\sqrt {p/q})= ((3/2)+(1/8p))1/\sqrt {3p}$$.

### MSC:

 11K38 Irregularities of distribution, discrepancy

### Keywords:

fractional part; discrepancy

Zbl 0938.11041
Full Text:

### References:

 [1] Baxa C.: On the discrepancy of the sequence $$(\alpha\sqrt{n})$$ II. Arch. Math) · Zbl 0905.11033 · doi:10.1007/s000130050208 [2] Baxa C., Schoißengeier J.: On the discrepancy of the sequence $$(\alpha\sqrt{n})$$. J. Lond. Math. Soc · Zbl 1009.11054 [3] Lerch M.: Sur quelques applications des sommes de Gauss. Ann. Mat. Pura Appl. (3. Ser.) 11 (1905), 79-91. [4] Schoißengeier J. : On the discrepancy of sequences $$(\alpha n^\sigma$$. Acta Math. Acad. Sci. Hung. 38 (1981), 29-43. · Zbl 0484.10032 · doi:10.1007/BF01917516
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