On the Class Numbers of Indefinite Binary Quadratic Forms with Discriminant dp<sup>2</sup>

Golubeva, E. P.

doi:10.1023/b:joth.0000035234.38002.1f

Cited by 1 publication

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“…See also [Pol86]. Other works, mostly related to the length of the period, which the reader might find related, may be found for example in many of the papers of Golubeva (such as [Gol02]) and in [Gri98], [BL05][MF93], [CZ04], [Coh77], [Hic73], [Kei]. Standing out in this context is the recent paper of McMullen which provides examples of sequences of quadratic irrationals in a fixed quadratic field with uniformly bounded c.f.e digits [McM09].…”

Section: References To Existing Resultsmentioning

confidence: 99%

On the evolution of continued fractions in a fixed quadratic field

Aka

Shapira

2018

JAMA

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We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the 'normal' statistics given by the Gauss-Kuzmin measure. As a by-product, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of k n α equals ck n +o(k (1− 1 16 )n ) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces. 1 We shall completely ignore the rational numbers, which correspond to finite sequences as well as real numbers outside the unit interval, for which an additional integer digit a 0 is needed.2 This correspondence is in fact a homeomorphism when N N is considered with the product topology.

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