Three Distance Theorem: Liang’s Proof: Example

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Example

The below ﬁgure shows the example of α = 7∕23 and N = 11. All multiples of 1/23 in interval [0,1] have been marked. Numbers 1,2,3,… below the interval line [0,1] indicate points {α},{2α},{3α},… .

Figure 1: α = 7∕23 and N = 11

PICT

Below is shown the repetitive reducing of all gaps, where each arrow indicates a reduce operation. Note that all reduce sequences are terminating at gaps (0,10) or (3,0), or region (1,8).

(1,11) → (0,10)

(11,8) → (10,7) → (9,6) → (8,5) → (7,4) → (6,3) → (5,2) → (4,1) →
(3,0)

(2,9) → (1,8)

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References

[1] Nitin Verma. Three Distance Theorem.
https://mathsanew.com/articles/three_distance_theorem.pdf.

[2] F. M. Liang. A Short Proof of the 3d Distance Theorem. Discrete Math., Vol 28 (3) (1979), 325–326.
https://doi.org/10.1016/0012-365X(79)90140-7.

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