Three Distance Theorem: Three Distance Theorem

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Three Distance Theorem

Let α < 1 be any positive real number and N any positive integer. As said earlier, we assume that N < q whenever α is rational p∕q. Now consider the increasing order of the N values {α},{2α},…,{Nα}, which must be all distinct (as concluded on page §). Say u₁,u₂,…,u_N denote this order such that {u₁,u₂,…,u_N} = {1,2,…,N} and, {uiα} < {ui+1α} for i = 1,2,…,N − 1. So the lowest and highest values are {u1α} and {uNα} respectively.

The basic statement of the theorem is as below.

Theorem 1 (Three Distance Theorem). For all i = 1,2,…,N − 1,

( |{ u1, 1 ≤ui< N + 1− u1 (a) u − u = u − u , N + 1 − u ≤u < u (b) i+1 i |( 1 N 1 i N − uN, uN ≤ui< N + 1 (c)

The points {uiα} and {ui+1α} are neighbors. The theorem says that for any pair of integers ui and ui+1, their diﬀerence can only be one among the three values: u1,u1 − uN,−uN.

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