Vol 213, No 9 (2022)

It is known that mappings occurring in quasiconformal analysis can be defined in several equivalent ways: 1) as homeomorphisms inducing bounded composition operators between Sobolev spaces; 2) as Sobolev-class homeomorphisms with bounded distortion whose operator distortion function is integrable; 3) as homeomorphism changing the capacity of the image of a condenser in a controllable way in terms of the weighted capacity of the condenser in the source space; 4) as homeomorphism changing the modulus of the image of a family of curves in a controllable way in terms of the weighted modulus of the family of curves in the source space. A certain set function, defined on open subsets, can be associated with each of these definitions. The main result consists in the fact that all these set functions coincide. Bibliography: 48 titles.

Let $a_1, …, a_s$ be integers and $N$ be a positive integer. Korobov (1959) and Hlawka (1962) proposed to use the points$$x^{(k)}=(\{\frac{a_1 k}N\}, …, \{\frac{a_1 k}N\}),\qquad k=1,…, N,$$as nodes of multidimensional quadrature formulae. We obtain some new results related to the distribution of the sequence $K_N(a)=\{x^{(1)},…,x^{(N)}\}$. In particular, we prove that$$\frac{\ln^{s-1} N}{N \ln\ln N} \underset{s}\ll D(K_N(a))\underset{s}\ll \frac{\ln^{s-1} N}{N} \ln\ln N$$for ‘almost all’ $a\in (\mathbb Z_N^*)^s$, where $D(K_N(a))$ is the discrepancy of the sequence $K_N(a)$ from the uniform distribution and $\mathbb Z^*_N$ is the reduced system of residues modulo $N$.Bibliography: 18 titles.

We show that palindromic continued fractions exist in an arbitrary dimension. For dimension $n=4$ we also prove a criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry. Klein polyhedra are considered as multidimensional generalizations of continued fractions. Bibliography: 11 titles.