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The Derivative of Minkowski's ?(x) Function

✍ Scribed by J. Paradı́s; P. Viader; L. Bibiloni

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
124 KB
Volume
253
Category
Article
ISSN
0022-247X

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✦ Synopsis

Minkowski's ? x function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it enables us to prove that its derivative, when it exists in a wide sense, can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than Ž . Ž . k s 5.31972, and ?Ј x exists, then ?Ј x s 0. In the same way, if the same average Ž . is less than k s 2 log ⌽, where ⌽ is the golden ratio, then ?Ј x s ϱ. Finally 2 some results are presented concerning metric properties of continued fractions and alternated dyadic expansions.

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