TY - JOUR
T1 - On the evolution of continued fractions in a fixed quadratic field
AU - Aka, Menny
AU - Shapira, Uri
N1 - Funding Information: ∗M. A. acknowledges the support of the Advanced research Grant 226135 from the European Research Council, the ISEF foundation, the Ilan and Asaf Ramon memorial foundation, and the Hoffman Leadership and Responsibility fellowship program at the Hebrew University of Jerusalem. †U. S. acknowledges the support of the Advanced research Grant 228304 from the European Research Council. 1We 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 a0 is needed. 2This correspondence is in fact a homeomorphism when NN is considered with the product topology.
PY - 2018/2/1
Y1 - 2018/2/1
N2 - 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 byproduct, 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 knα equals ckn + o(k15n/16) 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.
AB - 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 byproduct, 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 knα equals ckn + o(k15n/16) 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.
UR - http://www.scopus.com/inward/record.url?scp=85042912535&partnerID=8YFLogxK
U2 - 10.1007/s11854-018-0012-4
DO - 10.1007/s11854-018-0012-4
M3 - مقالة
SN - 0021-7670
VL - 134
SP - 335
EP - 397
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 1
ER -