Continued fractions of quadratic irrationals
Thursday May 3, 2007
02:00PM - 03:00PM
Speakers:
Vladimir Arnold
Abstract:
The continued fraction of the root x of the equation x^2+x=84 has thelength 21 period [1,2,8,1,5,4,2,2,1,1,1,1,2,2,4,5,1,8,2,1,17 ] which is palindromically symmetric (read it backwards).
The talk will discuss the pecularities of the periods of continued fractions of the roots of quadratic equations x^2+px+q=0. The length T(p,q) of this period has some turbulent growth rate for growing values of the integral coefficients p and q, and its averaged behavior will be described (as well as some other turbulent arithmetics results).
A year ago V. Bykovsky and his Vladivostok students have proved Arnold's old conjecture on the asymptotical Gauss-Kuz'min statistics of the distribution of the T elements of such periods. They all have the same palindromic property, that had been proved by Arnold for continued fractions of the square roots of rational numbers (and is visible above).
Extensions of these theories to higher-dimensional continued fractions to algebraic numbers of higher degree will be also discussed, leading to unsolved algorithmical decidability problems on the topological classification of the 2-torus' triangulations, related to cubical algebraic number fields.
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