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Eleven Euclidean distances are enough. (English) Zbl 1145.11053

It is a well known fact that while the sequence of fractional parts of integer multiples of a rational number is periodic, the same sequence for an irrational number is uniformly distributed modulo 1. A classical result in the study of such sequences is the three-distance theorem, which states that the set of distinct distances between consecutive elements of the sequence of fractional parts of the first \(n\) integer multiples of a real number has cardinality at most three for any \(n\). The main objective of the paper under review is to provide a generalization of the statement about finiteness of this set of distinct distances to higher dimensions under a suitable interpretation.
Sequences of fractional parts of integer multiples of a real number are often thought of as arising out of rotations on a circle of unit circumference. Then the three-distance theorem can be thought of as a statement about champions in a tournament. The players in the tournament are geodesics between pairs of points on the circle corresponding to different fractional parts. Two edges play each other if and only if they overlap, and an edge loses only against edges of shorter length that it plays against. According to the three-distance theorem, there are at most three distinct values for the lengths of undefeated edges.
In the paper under review, the author considers fractional parts of multiples of a vector of real numbers in the plane and in higher dimensions, and two edges play each other if their projections along any axis overlap. The author proves that in the plane, there are at most 11 values for the lengths of undefeated edges, although he remarks that numerical evidence suggests that the true value could be as small as 3. He also proves that the number of distinct distances in three dimensions is at most 74.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
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References:

[1] Chevallier, N., Geometrie des suites de Kronecker, Manuscripta Math., 94, 231-241 (1997) · Zbl 0893.11028
[2] Chung, F. R.K.; Graham, R. L., On the set of distances determined by the union of arithmetic progressions, Ars Combin., 1, 57-76 (1976) · Zbl 0352.10006
[3] Geelen, J. F.; Simpson, R. J., A two-dimensional Steinhaus theorem, Australas. J. Combin., 8, 169-197 (1993) · Zbl 0804.11020
[4] Liang, F. M., A short proof of the \(3d\) distance theorem, Discrete Math., 28, 325-326 (1979) · Zbl 0427.05014
[5] Sós, V. T., On the distribution mod 1 of the sequence \(\{n \alpha \}\), Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 1, 127-134 (1958) · Zbl 0094.02903
[6] Świerczkowski, S., On successive settings of an arc on the circumference of a circle, Fund. Math., 46, 187-189 (1958) · Zbl 0085.27203
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