Siegel measures
The results of the present paper represent a middle ground of sorts between the general Chebyshev theorem of Mazur and the restricted prime geodesic theorem of Veech. Let \(G_N=SL(N,\mathbb{R})\) and \(T_n=SL(N,\mathbb{Z})\). Equip \(G_N/T_N\) with its normalised Haar measure \(\mu_N\). If \(\psi\geq 0\) is a Borel function on \(R^N\) define \(\widehat\psi\) by \(\widehat\psi(gT_N)=\sum_{\vartheta\in\mathbb{Z}^N\setminus\{0\}}\psi(g\cdot\vartheta)\). Define \(M_N\) to be the set of Borel measures \(\upsilon\) on \(R^N\) such that \(M(\upsilon)<\infty\) and \(N_\nu(R)=\upsilon(B(0,R))\). The main theorem for Siegel measures is: If \(\mu\) is a Siegel measure, then there exists a constant \(c(\mu)<\infty\) such that: (i) If \(\psi\geq 0\) is a Borel function, then \(\int_{M_N}\widehat\psi(\upsilon)\mu(d\upsilon)=c(\mu)\int_{\mathbb{R}_N}\psi(u)du\). (ii) If \(\sigma_N\) is the area of the unit sphere in \(\mathbb{R}^N\), then \(\lim_{R\to\infty}\|{N_\nu(R)\over R^N}-{c(\mu)\sigma_N\over N}\|_1=0\). (iii) If \(\mu\) is supported on \(M_N^e=\{\upsilon\in M_N\mid\upsilon(-U)=\upsilon(U)\), \(U\) Borel\} then for all \(\psi\in C_c(R^N)\) \[ \lim_{R\to\infty}\left\|{1\over R^N}\int_{\mathbb{R}^N}\psi\left({\upsilon\over R}\right)\nu(d\upsilon)-C(\mu)\int_{\mathbb{R}^N}\psi(u)du\right\|_1=0. \] If \(N>2\), and if \(\mu\) is such that \(\widehat\psi\in L^2(\mu)\) for all \(\psi\in C_c(\mathbb{R}^N)\) then convergence in (ii) and (iii) also holds pointwise a.e. \(\mu\). As a consequence the main result for quadratic differentials is obtained. The restricted prime geodesic theorem of Veech along with a consequence is proved using the equidistribution theorem of \textit{A. Eskin} and \textit{C. McMullen} [Duke Math. J. 181-209 (1993; Zbl 0798.11025)]. It is also proved that if \(\mu\) is a Siegel measure, there exists \(C\geq 0\) and a singular Siegel measure \(\mu^s\) such that \(\mu=S^C*\mu^s\), where \(S^C:M_N\to M_N\) is defined by \(S^C\upsilon=\upsilon+cm\) and \(m=m(du)\) is the Lebesgue measure on \(R^N\). While the focus in this paper has been on periodic trajectories, entirely analogous results follow by the same technique for sets in the plane which represent simple geodesics joining cone points for the metrics \(| w|^2\), \(w\) a holomorphic 1-form.
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