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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

Citations:

Zbl 0938.11041
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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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