On the gap structure of sequences of points on a circle

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 x k =(log 2 (2 k −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.