Irregularities in the Distributions of Finite Sequences

Suppose (x1, x1 ,..., xs+J is a sequence of numbers with xi E [0, 1) which has the property that for each r < s and for each k < r, the subinterval [k/r, (k + l/n)) contains at least one point of the subsequence (xl, xz ,..., x7*). For fixed d, we wish to find the maximum s = s(d) for which such a sequence exists. We show that s(d) < 4(d+a12 for all d and that s(O) = 17. Let X = (x1, x2 ,...) be a sequence of points in the interval [0, l), Z a subinterval, j Z j its length and X,(Z) the number of x, in Z with m < n. Let ZQX) be the least upper bound of j X,(Z)-n / Z I I for Z varying in [0, 1). Z?,(X) was first proved to be unbounded by van Ardenne-Ehrenfest [2], [3] who showed (settling a conjecture of van der Corput [l]) F,(X) > c, log log n/log log log II. (1) This was later improved by K. F. Roth [4] who established t;l?(X) > c2 d'log. In this note we consider the following finite variant of this problem: For n 3 1,0 < k < n, define B I k k+l n,k = y ,-. n 1 Fix an integer d 3 0 and suppose (x1 , xg ,..., x,+,J is a sequence with xi E [0, 1) and with s = s(d) chosen to be maximal such that for each r Q s and each k < r, BrPk contains at least one point of the subsequence (x1 3 x2 ,-**2 &+a). The fact that s(d) < co follows from (2). In fact, the results of [4] can be used to show s(d) < 2216d* for sufficiently large d.