Diophantine properties of continued fractions on the Heisenberg group (Q2789383)

Let \(S\) be a special Heisenberg group. Denote \(\| . \|\) the Cygan-Koranyi norm and \(S(\mathbb Z)\) the integer points of \(S\). Set \(\operatorname{rad}(X)=\sup \{ \| h\|; h\in X\}\) and \(R_X=\prod_{n=1}^\infty (1+\operatorname{rad}(X)^n)^2\). For two points \(h_1=(u_1,v_1)\) and \(h_2=(u_2,v_2)\) with \(h_1,h_2\in S\) let us define the distance \(d(h_1,h_2)=| \overline{v_1}-\overline{u_1}u_2+v_2|\). Let \(K\) be the fundamental domain for \(S\) under the action of left translation by \(S(\mathbb Z)\), \(\operatorname{rad}(K)<1\) and the boundary of \(K\) is piece-wise smooth. Let \(h\in S\setminus S(\mathbb Q)\) be an irrational point with \(n\)th convergent \((\frac {r_n}{q_n},\frac {p_n}{q_n})\). Let \((\frac RQ,\frac PQ)\) be a different rational point in \(S\) in lowest terms. The author proved that if \(| Q| <\frac 1{(2 \operatorname{rad} K R_K)^2}| q_n|\) then \(d((\frac RQ,\frac PQ),h)>d((\frac {r_n}{q_n},\frac {p_n}{q_n}),h)\). He also proved several other analogous results as Hurwitz's theorem.