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Do algebraic numbers follow Khinchin's Law?

Autor: Philipp Sibbertsen and Timm Lampert and Karsten Müller and Michael Taktikos
Nummer: 686, May 2021, pp. 14
JEL-Class: -. MR-Class: 11J68; 11A55; 11J70; 11K45; 11K60; 65C20; 62-08

This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree >2 separately. For random numbers the coefficients are distributed by the Gauss-Kuzmin distribution (also called Khinchin's law). We apply the Kullback Leibler Divergence (KLD) to show that the Gauss-Kuzmin distribution does not fit well for algebraic numbers of degree > 2. Our suggestion to truncate the Gauss-Kuzmin distribution for finite parts fits slightly better, but its KLD is still much larger than the KLD of a random number. We consider differences regarding Khinchin's constant and Khinchin's approximation speed between random and algebraic numbers and conclude that laws concerning the random numbers do not automatically carry over to the algebraic numbers.


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