Do algebraic numbers follow Khinchin's Law?

TL;DR

This paper investigates whether algebraic numbers of degree greater than two follow Khinchin's Law, using numerical analysis and divergence measures, and finds no strong evidence supporting such a behavior.

Contribution

The study provides the first detailed numerical analysis suggesting algebraic numbers do not follow Khinchin's Law, challenging common assumptions based on measure-theoretic results.

Findings

KLD shows poor fit of Gauss-Kuzmin distribution for algebraic numbers

Truncated distributions fit slightly better but still poorly

No evidence supports applying Khinchin's theorems to algebraic numbers

Abstract

The coefficients of the regular continued fraction for random numbers are distributed by the Gauss-Kuzmin distribution according to Khinchin's law. Their geometric mean converges to Khinchin's constant and their rational approximation speed is Khinchin's speed. It is an open question whether these theorems also apply to algebraic numbers of degree $>2$. Since they apply to almost all numbers it is, however, commonly inferred that it is most likely that non quadratic algebraic numbers also do so. We argue that this inference is not well grounded. There is strong numerical evidence that Khinchin's speed is too fast. For Khinchin's law and Khinchin's constant the numerical evidence is unclear. 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. So, if it converges the convergence is non uniform and each algebraic number has its own bound. We conclude that there is no evidence to apply the theorems that hold for random numbers to algebraic numbers.