Vol 211, No 5 (2020)

All simple closed geodesics on regular tetrahedra in Lobachevsky space are described. The asymptotic behaviour is found for the number of simple closed geodesics of length $\leqslant L$, as $L$ tends to infinity. Bibliography: 22 titles.

Let $\Gamma$ be a rank-$s$ lattice in $\mathbb R^s$. The convex hulls of the nonzero lattice points lying in orthants are called the Klein polyhedra of $\Gamma$. This construction was introduced by Klein in 1895, in connection with generalizing the classical continued-fraction algorithm to the multidimensional case. Arnold stated a number of problems on the statistical and geometric properties of Klein polyhedra. In two dimensions the corresponding results follow from the theory of continued fractions. An asymptotic formula for the mean value of the $f$-vectors (the numbers of facets, edges and vertices) of 3D Klein polyhedra is derived. This mean value is taken over the Klein polyhedra of integer 3D lattices with determinants in $[1,R]$, where $R$ is an increasing parameter. Bibliography: 27 titles.

Let $\mathbb N_0$ be the set of positive integers whose binary decompositions contain an even number of ones. We give a bound for the trigonometric sum of special form over numbers in $\mathbb N_0$; using this bound, we derive an asymptotic formula for the number of solutions to Waring's equation in positive integers in $\mathbb N_0$, and also a bound for the number of terms in the last equation, which is sufficient for the equation to be solvable in integers in $\mathbb N_0$. Bibliography: 9 titles.