On the gap structure of sequences of points on a circle |
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Affiliation: | Computer Science Department, Stanford University, Stanford, California 94305, USA |
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Abstract: | Considerable mathematical effort has gone into studying sequences of points in the interval [0, 1) which are evenly distributed, in the sense that certain intervals contain roughly the correct percentages of the first n points. This paper explores the related notion in which a sequence is evenly distributed if its first n points split a given circle into intervals which are roughly equal in length, regardless of their relative positions. The sequence xk=(log2 (2k−1) mod 1) was introduced in this context by De Bruijn and Erdös. We will see that the gap structure of this sequence is uniquely optimal in a certain sense, and optimal under a wide class of measures. |
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