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The discrepancy of \((n\alpha)_{n\geq 1}\). (English) Zbl 0786.11043

For a real irrational \(\alpha\) and a positive integer \(N\) let \[ \omega_ N^ += \sup_{0\leq x\leq 1} \bigl( \sum_{n=1}^ N c_{[0,x)}(\{n\alpha\})- Nx\bigr) \quad\text{and}\quad \omega_ N^ - (\alpha)= \sum_{0\leq x\leq 1} \bigl( Nx- \sum_{n=1}^ N c_{[0,x)}(\{n\alpha\})\bigr). \] Here \(c_ M\) denotes the characteristic function of the set \(M\) and \(\{y\}= y-[y]\) is the fractional part of the real \(y\). \(D_ N^*(\alpha)= \max(\omega_ N^ + (\alpha),\omega_ N^ - (\alpha))\) is known as the discrepancy of \((n\alpha)_{n\geq 1}\). Since the beginning of the twenties one looks for estimates for \(D_ N^*(\alpha)\) from above, at least for certain classes of \(\alpha\)’s. See the papers in the Abh. Math. Semin. Hamburg of H. Behnke [ibid. 1, 252-267 (1922; JFM 48.0186.01), ibid. 3, 261-318 (1924; JFM 50.0124.03)], G. H. Hardy and J. E. Littlewood [ibid. 1, 212-249 (1922; JFM 48.0187.01)], E. Hecke [ibid. 1, 54-76 (1922; JFM 48.0184.02)] and of A. Ostrowski [ibid. 1, 77-98 (1922; JFM 48.0185.01)].
In this paper the author gives for all irrational \(\alpha\) best possible estimates in the following form: let \(\alpha= [0;a_ 1,a_ 2,\dots]\) be the continued fraction expansion of \(\alpha\) with convergents \(p_ n/q_ n\). For \(i,j\geq 0\) let \[ s_{ij}= q_{\min(i,j)} (q_{\max(i,j)} \alpha-p_{\max(i,j)}) \qquad\text{and}\qquad \varepsilon_ i= {\textstyle {1\over 2}}((-1)^{a_{i+1}}-1) \prod_{\textstyle{{j\leq i} \atop {2\mid j-i}}} (-1)^{a_{j+1}}. \] Then for \(m\to\infty\): \[ \begin{aligned} 4 \max_{1\leq N<q_{m+1}} \omega_ N^ +(\alpha) &= \sum_{2\mid i\leq m} a_{i+1}+ \sum_{2\mid i\leq m} \sum_{2\mid j\leq m} \varepsilon_ i \varepsilon_ j s_{ij}+ O(1) \qquad\text{ and}\\ 4\max_{1\leq N<q_{m+1}} \omega_ N^ - (\alpha) &= \sum_{2\nmid i\leq m} a_{i+1}- \sum_{2\nmid i\leq m} \sum_{2\nmid j\leq m} \varepsilon_ i \varepsilon_ j s_{ij}+ O(1).\end{aligned} \] The \(O\)-constants are absolute. The author deduces for numbers \(\alpha\) of bounded density (i.e. if \(\sum_{i\leq m} a_ i=O(m)\)) and especially for all quadratic irrationals \(\alpha\) formulas for \(\varlimsup_{N\to\infty} D_ N^*/\log N\) in terms of the continued fraction expansion of \(\alpha\). The paper ends with best possible estimates for \(D_ N^*(\alpha)\) for numbers of the form \(\text{coth }{1\over t}\), \(t\geq 1\) an integer, and for \(e\).

MSC:

11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
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References:

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