A Geometrical Proof of a Theorem of Hurwitz

Lester R. Ford

doi:10.1017/S0013091500029692

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In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequality

how small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?

References

page 59 note * Mathematische Annalen 39 (1891), 279–84.CrossRef Google Scholar

The problem was also solved by Borel, , Journal de Mathématiques, 5th Ser., Vol. 9 (1903), 329–.Google Scholar

Since the present paper was read to the Society, there has come to hand the current issue of the Journal de Mathématiques, which contains a simple proof of the theorem by Humbert.

page 61 note * Humbert has shown that the coordinates of these peaks are the fractions of Hermite mentioned in Section 1. Journal de Mathématiques, 7th Ser., Vol. 2 (1916), 79–103.Google Scholar

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A Geometrical Proof of a Theorem of Hurwitz

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