Continued fractions on the Heisenberg group

Abstract

Abstract. We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued fractions. We then discuss dynamical properties of the associated Gauss map, comparing them with base-b expansions on the Heisenberg group and continued fractions on the complex plane.

Intrinsic diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group

“…The Heisenberg continued fraction with digits {γ i } is given by K{γ i } = lim n→∞ γ 0 * ι(γ 1 * ι(γ 2 * · · · * ι(γ n ))), if this limit exists. We showed in [18] that every point h ∈ S 1 admits a continued fraction expansion, and characterized rational points in S 1 as exactly those points with a finite expansion.…”

“…The continued fraction digits of h are found in [18] by repeatedly applying a generalized Gauss map T to the point h. More explicitly, let K be the Dirichlet region for S 1 (Z) centered at the origin, i.e., K is the set of points closer to 0 than to any other point in The convergents of a continued fraction can be canonically written as rational numbers, and we write for each n: r n q n , p n q n = γ 0 * ι(γ 1 * ι(γ 2 * · · · * ι(γ n ))).…”

“…Consider now a horoball B h based at h = h 0 = ι(γ 1 * ι(γ 2 · · · ι(γ n * 0) · · · )) with unknown horoheight s. Interpreting Heisenberg left multiplication and ι as elements of Γ as in [18], we have B h = ι(γ 1 * ι(γ 2 · · · ι(γ n * B 0 ) · · · )).…”

Lagrange's theorem for continued fractions on the Heisenberg group

“…If h is an irrational point (that is, not in Q[i] 2 ), then we have that h has an infinite number of continued fraction digits with respect to Dirichlet domain and that K{γ i } ∞ i=0 exists and equals h. (In other words, the continued fraction digits generated in this way do, in fact, form a continued fraction expansion for the original point.) If h is rational (that is, in Q[i] 2 ), then it has a finite number of continued fraction digits, say m of them, with respect to Dirichlet domain and h = K{γ i } m i=0 (see [6,Theorem 1.3]). For the Dirichlet domain K D , the map T acting on the intersection of K D and the set…”

“…Classical real continued fractions are deeply connected with the study of geodesics in onedimensional complex hyperbolic space. Motivated by a desire to study higher-dimensional complex hyperbolic space, the author and Anton Lukyanenko began studying continued fractions on the Heisenberg group [6,10]. We consider the Heisenberg group in its Siegel model, given by the space…”