Abstract
For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for whichthe step problem: for the setrearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.
Publisher
Cambridge University Press (CUP)
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