The lattice point problem in the hyperbolic plane (Q2736202)

Let \(\Gamma\) be a geometrically finite torsionfree Fuchsian group acting on the hyperbolic plane with exponent of convergence \(\delta>\frac 12\), discrete eigenvalues \(\lambda_1,\lambda_2,\dots,\lambda_n\in[0,\frac 14[\) and corresponding normalized real eigenfunctions \(\varphi_1,\dots,\varphi_n\). The author studies the asymptotic behaviour of the counting function NEWLINE\[NEWLINEN(R;z,w):=\#\{\gamma\in\Gamma:\cosh d(\gamma z,w)\leq R\}NEWLINE\]NEWLINE of the lattice points \(\gamma z\) inside a hyperbolic disc, where \(d\) denotes the hyperbolic distance. The main theorem of the paper under review states:NEWLINENEWLINENEWLINELet \(s_k:=\frac 12+\sqrt{\frac 14-\lambda_k}\) \((k=1,\dots,n)\) where \(s_1=\delta\). Then, as \(R\to\infty\), NEWLINE\[NEWLINEN(R;z,w)=\sum_k 2\sqrt{\pi} \frac{\Gamma(s_k-\frac 12)}{\Gamma(s_k+1)} \varphi_k(z)\varphi_k(w)R^{s_k}+O(R^{\frac 13(1+\delta)}),NEWLINE\]NEWLINE where the sum extends over all \(k\) such that \(\frac 13(1+\delta)<s_k\leq 1\).NEWLINENEWLINENEWLINEAs the author points out, this improves on the estimate of \textit{P. Lax} and \textit{R. S. Phillips} [J. Funct. Anal. 46, 280-350 (1982; Zbl 0497.30036))] whose estimate involves an additional factor \((\log R)^{5/6}\) in the error term. (However, the author seems to be unaware of the fact that this logarithmic factor was already removed by \textit{B. M. Levitan} [Russ. Math. Surv. 42, No.~3, 13-42 (1987; Zbl 0632.10048)]. Levitan considers cofinite groups (see p. 15), but he also points out that his results extend to geometrically finite groups.) The author's proof is based on estimates of the counting function from above and from below and on Maurin's version of spectral theory.NEWLINENEWLINENEWLINEUnder slightly more restrictive hypotheses, the author also gives explicit estimates for \(N(R;z,w)\) from above and from below. This enables him to approximate the exponent of convergence \(\delta\) by means of the counting function. The method is carried out numerically in the case of Hecke groups.NEWLINENEWLINENEWLINEThere are also rather detailed historical remarks, but these are not always sufficiently complete and sometimes misleading. It is (correctly) stated on p. 57 that the reviewer proved the monotonicity of the abscissa of convergence for the Hecke groups \(G(\lambda)\) \((\lambda>2)\), but this was never published and the proof is not contained in the reference quoted.