Ergodic theory on homogeneous spaces and the calculation of lattice points in polyhedrons (Q5934146)

Let \(\Gamma\subset\mathbb{R}^d\) be a unimodular lattice and \(O\subset\mathbb{R}^d\) be a compact domain with a piecewise smooth boundary. The problem of calculating the number of lattice points in a dilating domain by evaluating the remainder \(R(tO,\Gamma)\) as \(t\to\infty\) is a classical problem of the geometry of numbers [see \textit{P. Gruber} and \textit{C. G. Lekkerkerker}, Geometry of Numbers, 2nd ed. (1987; Zbl 0611.10017)]. From the work of Skriganov it is found that for certain \(O\) and \(\Gamma\), the remainder \(R(tO,\Gamma)\) can be logarithmically small. The basic result in this paper is as follows: Assume that \(P\subset \mathbb{R}^d\) is an arbitrary polyhedron. Then for almost all \(\Gamma\) in SL\((d, \mathbb{R})/\text{SL}(d,\mathbb{Z})\), in the sense of an invariant probability measure \(\mu\) on SL\( (d, \mathbb{R})/\text{SL}(d,\mathbb{Z})\), the relation \[ R(tP,\Gamma)\ll(\log t)^{d-I+ \varepsilon} \] is valid with an arbitrary small \(\varepsilon>0\). Thus from the metric point of view, the appearance of a logarithmically small remainder is quite typical for polyhedrons. IT is established that such anomalies in the behaviour of remainders are related with ergodic properties of certain flows on homogeneous spaces.