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On the measure of large trigonometric sums. (English) Zbl 0643.10031
Collective phenomena, Rep. Moscow Refusnik Semin., 6th Int. Conf., Stockholm 1983, 7th Int. Conf., Tel Aviv 1984, Ann. N. Y. Acad. Sci. 452, 363-371 (1985).
[For the entire collection see Zbl 0627.00021.]
Let \(\mu (u)=\max_{K} \mu_ K(u)\), where \(\mu_ K(u)\) is the measure of the set of all values of \(\alpha\) for which \[ | S_ K(\alpha)| =| \sum^{k-1}_{j=0}e(\alpha a_ j)| \geq k-u, \] \(K=\{a_ 0<a_ 1<....<a_{k-1}\}\) and \(a_ j\in {\mathbb{Z}}\), \(j=0,1,...,k-1\). It is shown in the paper that for every k it is possible to construct a sufficiently small \(\epsilon >0\) such that for \(u<\epsilon k\) we have \[ \mu (u)=2\alpha_ 0=\sqrt{6}/(\pi k)\sqrt{u/k}(1- 3u/(10k)+...), \] \(\alpha_ 0\) is the smallest \(\alpha >0\) satisfying \(\sin (\pi \alpha k)/\sin (\pi \alpha)=k-u.\) Some results about the structure of the set \(E_ K(u)=\{\alpha: \alpha \in [0,1]\), \(| S_ K(\alpha)| \geq k-u\}\) are also obtained.
Reviewer: G.Kolesnik
11L03 Trigonometric and exponential sums, general