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**Distribution of points on arcs.**
*(English)*
Zbl 1156.11310

Let \(z_1,\dots, z_N\) be complex numbers situated on the unit circle \(\mathbb U:=\{z\in \mathbb C\mid |z| = 1\}\), and write \(S := z_1 +\cdots+ z_N\). In 1962 G. A. Freiman [Izv. Vyssh. Uchebn. Zaved., Mat. 1962, No. 6(31), 131–144 (1962; Zbl 0171.00803)] established a lemma showing that if \(z_1,\dots, z_N\in \mathbb U\) are “uniformly distributed on arcs of length \(\pi\)”, then the sum \(z_1 +\cdots+ z_N\) is small in absolute value, \(S\leq 2n-N\). The assumption “any open arc of \(\mathbb U\) of length \(\pi\) contains at most \(n\) of the numbers \(z_1,\dots, z_N\)” implies readily that \(N\leq 2n\).

In practice, an analog of Lemma 1 is needed with the arcs of length \(\pi\) replaced by arcs of other prescribed lengths; see for instance [B. Green and I. Z. Ruzsa, Bull. Lond. Math. Soc. 38, No. 1, 43–52 (2006; Zbl 1155.11307), T. Schoen, Integers 3, Paper A17, 6 p., electronic only (2003; Zbl 1089.11009)]. The author generalizes Freiman’s result as follows:

Theorem 1: Suppose that any open arc of length \(\phi\in (0, \pi]\) of the unit circle contains at most \(n\) of the numbers \(z_1,\dots, z_N\). Then

\[ |S|\leq 2n-N+2(N-n) \cos(\phi/2). \]

Theorem 1 is sharp, at least in the range \(N/2\leq n \leq N\): the bound is attained, for instance, if \(2n-N\) of the numbers \(z_j\) equal one, \(N-n\) equal \(\exp(i\phi/2)\), and \(N-n\) equal \(\exp(-i\phi/2)\).

Theorem 2: Suppose that any open arc of length \(\pi\) of the unit circle contains at most \(n\) of the numbers \(z_1,\dots, z_N\) and suppose, in addition, that for any \(1\leq i< j\leq N\) the length of the (shortest) arc between \(z_i\) and \(z_j\) is at least \(\delta > 0.\) Then \[ |S|\leq\frac{\sin (n-N/2)\delta}{\sin \delta/2} \]

provided that \(n\delta\leq \pi\). Theorem 2 is also sharp.

Finally the author notices that from Theorem 1 one can deduce the following more general result.

Theorem 1’. Let \(\lambda\leq 1/2\) and \(\nu\) be positive real numbers and let \(\mu\) be a probabilistic measure on the torus group \(\mathbb R/\mathbb Z\). Suppose that \(\mu(I) \leq \nu\) for any open interval \(I \subseteq\mathbb R/\mathbb Z\) of length \(|I| = \lambda\). Then \[ \biggl|\int_{\mathbb R/\mathbb Z}\exp(it)\, d\mu\biggr| \leq 2\nu-1 + 2(1-\nu)\cos(\pi\lambda). \]

In practice, an analog of Lemma 1 is needed with the arcs of length \(\pi\) replaced by arcs of other prescribed lengths; see for instance [B. Green and I. Z. Ruzsa, Bull. Lond. Math. Soc. 38, No. 1, 43–52 (2006; Zbl 1155.11307), T. Schoen, Integers 3, Paper A17, 6 p., electronic only (2003; Zbl 1089.11009)]. The author generalizes Freiman’s result as follows:

Theorem 1: Suppose that any open arc of length \(\phi\in (0, \pi]\) of the unit circle contains at most \(n\) of the numbers \(z_1,\dots, z_N\). Then

\[ |S|\leq 2n-N+2(N-n) \cos(\phi/2). \]

Theorem 1 is sharp, at least in the range \(N/2\leq n \leq N\): the bound is attained, for instance, if \(2n-N\) of the numbers \(z_j\) equal one, \(N-n\) equal \(\exp(i\phi/2)\), and \(N-n\) equal \(\exp(-i\phi/2)\).

Theorem 2: Suppose that any open arc of length \(\pi\) of the unit circle contains at most \(n\) of the numbers \(z_1,\dots, z_N\) and suppose, in addition, that for any \(1\leq i< j\leq N\) the length of the (shortest) arc between \(z_i\) and \(z_j\) is at least \(\delta > 0.\) Then \[ |S|\leq\frac{\sin (n-N/2)\delta}{\sin \delta/2} \]

provided that \(n\delta\leq \pi\). Theorem 2 is also sharp.

Finally the author notices that from Theorem 1 one can deduce the following more general result.

Theorem 1’. Let \(\lambda\leq 1/2\) and \(\nu\) be positive real numbers and let \(\mu\) be a probabilistic measure on the torus group \(\mathbb R/\mathbb Z\). Suppose that \(\mu(I) \leq \nu\) for any open interval \(I \subseteq\mathbb R/\mathbb Z\) of length \(|I| = \lambda\). Then \[ \biggl|\int_{\mathbb R/\mathbb Z}\exp(it)\, d\mu\biggr| \leq 2\nu-1 + 2(1-\nu)\cos(\pi\lambda). \]

Reviewer: Olaf Ninnemann (Berlin)