## 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 $\biggl|\int_{\mathbb R/\mathbb Z}\exp(it)\, d\mu\biggr| \leq 2\nu-1 + 2(1-\nu)\cos(\pi\lambda).$

### MSC:

 11J71 Distribution modulo one 11K36 Well-distributed sequences and other variations

### Citations:

Zbl 0171.00803; Zbl 1155.11307; Zbl 1089.11009
Full Text: