Distribution of Points on Arcs
نویسنده
چکیده
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 the length of the (shortest) arc between zi and zj is at least δ > 0. Then |S| ≤ sin (n−N/2)δ sin δ/2 provided that nδ ≤ π. These estimates are sharp. Received: 10/21/03, Revised: 7/5/04, Accepted: 9/24/04, Published: 9/1/05
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