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Euclidean outer billiard on a regular polygon (that is not a triangle, square or a hexagon) has aperiodic points, i.e., points where all iterates of the outer billiard map are defined and yield pairwise distinct images. This result answers a question of R. Schwartz posed at ICM 2022., Comment: 41 pages, 8 figures

In a recent paper "Solvability of equations in elementary functions" [arXiv:1911.10409] the insolvability in elementary functions of equation $\tan(x) - x = a$ was proved. This work applies the same topological method to prove the insolvability of equation $x^x = a$., Comment: 6 pages

We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Parsin, Szpiro) with modularity results and Masser-W\"ustholz isogeny estimates. In this work we explicitly bound the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus. In the first part we define and study coarse Hilbert moduli schemes with their heights and branch loci. In the second part we establish the effective Shafarevich conjecture for abelian varieties $A$ over a number field $K$ such that $A_{\bar{K}}$ has CM or $A_{\bar{K}}$ is of GL2-type and isogenous to all its $G_\mathbb Q$-conjugates. In the third part we continue our explicit study of the Parsin construction given by the forgetful morphism of Hilbert moduli schemes. We now work out our strategy for arbitrary number fields $K$ and we explicitly bound the number of polarizations and module structures on abelian varieties over $K$ with real multiplications. In the last part we illustrate our results by applying them to two classical surfaces first studied by Clebsch (1871) and Klein (1873): We explicitly bound the Weil height and the number of their integral points., Comment: Comments are always very welcome

This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible questions that might obstruct our path to the correct answer and final solution. With this method, the desired configuration can be formulated. This method is effective for solving problems in Olympiads and mathematical competitions as well as for teaching combinations and counting science in the elementary school mathematics curriculum., Comment: 15 pages, 13 figures

Average distance between two points in a unit-volume body $K \subset \mathbb{R}^n$ tends to infinity as $n \to \infty$. However, for two small subsets of volume $\varepsilon > 0$ the situation is different. For unit-volume cubes and euclidean balls the largest distance is of order $\sqrt{-\ln \varepsilon}$, for simplexes and hyperoctahedrons $-$ of order $-\ln \varepsilon$, for $\ell_p$ balls with $p \in [1;2]$ $-$ of order $(-\ln \varepsilon)^{\frac{1}{p}}$. These estimates are not dependent on the dimensionality $n$. The goal of the paper is to study this phenomenon. Isoperimetric inequalities will play a key role in our approach.

We show that the tensor product of modules of tensor fields is a noetherian module as a module over any graded Lie subalgebra of finite codimension in the Lie algebra of polynomial vector fields on $\mathbb{R}^n$. As a corollary, we prove the conjecture of I.M.Gelfand announced at ICM'1970 at Nice on finite dimensionality of continuous cohomology of graded Lie subalgebras of formal vector fields $W_n$., Comment: preliminary draft, comments are welcome

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and utilizing the representation theory of the symmetric group. But this technique is unavailable in characteristic $p>0$. An alternative method, which succeeds, is the process of ``hiking'' a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial that contains Capelli polynomials, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper we develop hiking further to nonhomogeneous polynomials, to apply to the representability question. Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In the appendix we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems., Comment: 30 pp. arXiv admin note: substantial text overlap with arXiv:1805.04450

We prove that for an arbitrary $\varepsilon > 0$ holds \[ \chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where $\chi(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of monochromatic points at the distance 1 apart.

This is the third part of a cycle of papers devoted to the construction of a finitely presented infinite nil-semigroup satisfying the identity $x^9 = 0$. This construction answers the problem of L. N. Shevrin and M. V. Sapir, posed, for example, in the Sverdlovsk notebook. A semigroup is realized as a set of path encodings on a family of special uniformly elliptic complexes. In the first paper of the cycle Finitely defined nil semigroup: complexes with uniform ellipticity, a sequence of complexes was constructed with a set of geometric properties. In the second work of the series Deterministic Coloring of a Family of Complexes a finite letter encoding was introduced on the vertices and edges of the constructed complexes. The deterministic property of such a coloring was proved, which makes it possible to introduce a finite set of defining relations on the set of words-codings of paths on complexes. In this paper, we describe an algorithm for reducing an arbitrary semigroup word to canonical form. It is also proved that a word containing a subword with period 9 can be reduced to zero using defining relations. Word encodings corresponding to sufficiently long paths are not reduced to zero and do not change their length, that is, the introduced semigroup is infinite., Comment: in Russian. The work was carried out with the help of the Russian Science Foundation, Grant 22-11-00177. The first author is the winner of the Young Mathematics of Russia competition

The challenges of examining random partitions of space are a significant class of problems in the theory of geometric transformations. Richard Miles calculated moments of areas and perimeters of any order (including expectation) of the random division of space in 1972. In the paper we calculate whole distribution function of random divisions of plane by poisson line process. The idea is to interpret a random polygon as the evolution of a segment along a moving straight line. In the plane example, the issue connected with an infinite number of parameters is overcome by considering a secant line. We shall take into account the following tasks: {\textbf 1.} On the plane, a random set of straight lines is provided, all shifts are equally likely, and the distribution law is of the form $F(\varphi).$ What is the area distribution of the partition's components? {\textbf 2.} On the plane, a random set of points is marked. Each point $A$ has an associated area of attraction, which is the collection of points in the plane to which the point $A$ is the nearest of the designated ones. In the first problem, the density of moved sections adjacent to the line allows for the expression of the balancing ratio in kinetic form. Similarly, you can write the perimeters kinetic equations. We will demonstrate how to reduce these equations to the Riccati equation using the Laplace transformation in this paper., Comment: 26 pages, 2 figures, more detailed exposition and pictures added

The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra $A$ the image of a multilinear polynomial on $A$ is a vector space. In this paper we prove it for the algebra of octonions $\mathbb{O}$ over a field satisfying certain specified conditions (in particular, we prove it for quadratically closed field and for field $\mathbb{R}$). In fact, we prove that the image set must be either $\{0\}$, $F$, the space of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the corresponding Malcev algebra., Comment: 14 pages

It is well known that if there exists a finite set of convex bodies on the plane with non-overlapping interiors, then there is at least one "extremal" one among them, i.e., some one which can be continuously "taken away to the infinity" (outside a large ball containing all other bodies). In 3-space a phenomenon of self-interlocking structures takes place. A self-interlocking structure is such a set of three-dimensional convex bodies with non-overlapping interiors that any infinitesimal move of any of them is possible only as a part of the move of all bodies as a solid body. Previously known self-interlocking structures are based on configurations of cut cubes, tetrahedra, and octahedra. In the present paper we discover a principally new phenomenon of 2-dimensional self-interlocking structures: a family of 2-dimensional polygons in 3-space where no infinitesimal move of any piece is possible. (Infinitely thin) tiles are used to create {\em decahedra}, which, in turn, used to create columns, which turn out to be stable when we fix some two extreme tiles. Seemingly, our work is the first appearance of the structure which is stable if we fix just two tiles (and not all but one). Two-dimensional self-interlocking structures naturally lead to three-dimensional structures possessing the same properties.