ONTHE GAP STRUCTUREOF SEQUENCES OF POINTS ONA CIRCLE bY

Considerable mathematical effort has gone into studying sequences of points in the interval [O,l> 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 \ = (log2(2k-1) mod 1) was introduced in this context by DeBruijn and Erd&%s. 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. This research was supported in part by National Science Foundation grant MCS 72-03752 A03, by the Office of Naval Research contract NOOOlb76-c-0330, by IBM Corporation, and a National Science Foundation Graduate Fellowship, Reproduction in whole or in part is permitted for any purpose of the United States government. Consider sequences of points on the circumference of a circle of radius 1/2n , or equivalently in the unit interval [O,l) . Such a sequence is called uniformly distributed if the percentage of the first n points which lie in any fixed interval approaches the length of that interval as n tends to infinity; this concept has been studied extensively [4]. We can arrive at a different notion of even distribution by considering instead the lengths of the gaps between elements of the sequence. For each n , the first n points of any sequence divide the circle into n intervals, and we shall study those sequences which make these intervals roughly equal in length, regardless of the order in which they occur around the circle. Putting this another way, we will study strategies for successively breaking a unit stick into smaller and smaller fragments, while attempting to arrange that the n stick fragments present at time n are as nearly equal in length as t possible, for all n. More formally, let us define an n-state to be a multiset containing n nonnegative real numbers which sum to one; the elements of the n-state specify the lengths of the sticks present at time n . An n-state S is a legal predecessor of an (n+l) -state T if there exists a number x in S such that S-(x) 5 T . It follows that the multiset T (S-(x]) must consist of exactly two numbers whose sum is x ; that is, T arises from S by breaking a stick of length x into two nonnegative fragments. A-stickbreaking strategy is then an infinite sequence of states (Snjn>l t where sn is an n-state and a legal predecessor of n+l for each n . Every sequence of points on the circle defines a unique