Distribution of Points on Arcs

@inproceedings{Lev2005DistributionOP,
  title={Distribution of Points on Arcs},
  author={Vsevolod F. Lev},
  year={2005}
}
Let z1, . . . , zN be complex numbers situated on the unit circle |z| = 1, and write S := z1 + · · · + zN . Generalizing a well-known lemma by Freiman, we prove the following. (i) Suppose that any open arc of length φ ∈ (0, π] of the unit circle contains at most n of the numbers z1, . . . , zN . Then |S| ≤ 2n − N + 2(N − n) cos(φ/2). (ii) Suppose that any open arc of length π of the unit circle contains at most n of the numbers z1, . . . , zN and suppose, in addition, that for any 1 ≤ i < j ≤ N… CONTINUE READING