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Let α be any positive real number with α < 1, and N any positive integer. For any real number x, let {x} denote the fractional part of x, i.e. {x} = x−⌊x⌋. Consider the points {0α}(= 0),{1α},{2α},{3α},…,{Nα} in interval [0,1]. For simplicity, we will assume in this discussion that all these points are distinct; similar reasoning will apply if they start repeating after {iα} for some i. These points will divide the interval [0,1] in N + 1 non-empty intervals, which we will call “Gaps”.
The basic statement of Three Distance Theorem (also known as Three Gap Theorem and Steinhaus Conjecture) is as follows. Please refer the article titled Three Distance Theorem [1] for more insights into this theorem.
Theorem 1 (Three Distance Theorem). The gaps created (as described above) have only upto three distinct lengths.
F. M. Liang gave a simple and elegant proof for a generic version of this theorem [2]. In this article, we will see an elaboration of that proof for the theorem presented above.
Any region (not necessarily a gap) with endpoints {iα} and {jα}, {iα} < {jα}, will be denoted as (i,j) (i can be 0). A region from some endpoint {iα} to endpoint 1 will be denoted by a special notation (i,0), because a region ending at 1 can be seen as wrapping around and ending at 0.
If a region contains some point {kα}, excluding its own endpoints, we will say that the region “encloses” that point.
Note that a gap is also one of the regions just described, which does not enclose any point. So we will use the same notation to denote a gap.
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