Gaps and steps for the sequence n θ mod 1
Noel B. Slater · Mathematical Proceedings of the Cambridge Philosophical Society · 1967
For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts { r θ}, namely: the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which the step problem: for the set rearranged 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.