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Том 86, № 3 (2022)

Жылы: 2022
Мақалалар: 5
URL: https://journals.rcsi.science/1607-0046/issue/view/8356

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Ашық рұқсат
Рұқсат жабық

Рұқсат берілді
Рұқсат жабық

Тек жазылушылар үшін

Articles

On Jutila's integral in the circle problem

Korolev M., Popov D.

Аннотация

We study a ‘correlation’ function $\mathcal{K}_{P} = \mathcal{K}_{P}(T;H,U)$ of the error term$P(t)$ in the circle problem, that is, the integral of the product$P(t)P(t+U)$ over the interval $(T,T+H]$, $1 ė U, H ė T$. The case of small $U$, $1\le U\ll \sqrt{T}$, was in essence studied by Jutila in 1984.It turns out that, for all these $U$ and sufficiently large $H$,$\mathcal{K}_{P}$ attains its maximum possible value. In this paper we study the caseof ‘large’ $U$, $\sqrt{T}\ll U\le T$, when the behaviour of $\mathcal{K}_{P}$ becomes morecomplicated. In particular, we prove that the correlation function may be positive and negativeof maximally large modulus as well as having very small modulus on sets of values of $U$of positive measure.

Көрсету

Жасыру

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(3):3-46

3-46

(Rus)

(JATS XML)

The real Plücker–Klein map

Krasnov V.

Аннотация

We consider the generalized Plücker–Klein map from the set of all real marked biquadricsto the set of real Kummer varieties. We find a necessary and sufficient condition on a real markedbiquadric in order that the corresponding real Kimmer variety be isomorphic to the real Kummervariety induced by the real Jacobian of a double covering of the pencil of quadrics through thegiven biquadric. We also give a deformation classification of the real Plücker–Klein map.

Көрсету

Жасыру

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(3):47-104

47-104

(Rus)

(JATS XML)

Outer billiards outside regular polygons: tame case

Rukhovich P.

Аннотация

We consider the periodicity problem, that is, the existence of an aperiodic point andfullness of measure of the set of periodic points for outer billiards outside regular $n$-gons.The lattice cases$n=3,4,6$ are trivial: no aperiodic points exist and the set of periodic points is of full measure.The cases $n=5,10,8,12$ (and only these cases) are regarded as tame. The periodicityproblems were solved for $n=5$ in a breakthrough paper by Tabachnikov, who pioneered a renormalization-scheme method for studying the arising self-similar structures.The case $n=10$ is similar to $n=5$ and was studied earlier by the present author. The presentpaper is devoted to the remaining cases $n=8,12$. We establish the existence of an aperiodic orbitin outer billiards outside regular octagons and dodecagons and prove that almost all trajectoriesof these outer billiards are periodic. In the regular dodecagon case we give a rigorouscomputer-assisted proof. We establish equivalence between the outer billiards outsidea regular $n$-gon and a regular $n/2$-gon, where $n$ is even and $n/2$ is odd.Our investigation is based on Tabachnikov's renormalization scheme.

Көрсету

Жасыру

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(3):105-160

105-160

(Rus)

(JATS XML)

New estimates for short Kloosterman sums with weights

Semenova N.

Аннотация

In the paper we obtain a new bound for short Kloosterman sums modulo a primewith a weight. The derivation of the bound is based on Karatsuba'smethod (1993–1995) of estimating incomplete Kloosterman sums and on a modification of the method proposed by Bourgain and Garaev (2014).The theorems proved in the paper refine results obtained earlier byKorolev (2010).

Көрсету

Жасыру

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(3):161-186

161-186

(Rus)

(JATS XML)

On a convex polyhedron in a regular point system

Shtogrin M.

Аннотация

Faceting with a ‘filling’. An ideal crystal structure consists of finitely many equal and parallel translational point lattices. In $\mathbb R^3$ it extends unboundedly in all directions. We distinguish in it a finite part situated in a closed convex polyhedron every face of which contains nodes of a translational point lattice involved in the structure not belonging to the same straight line. Such a polyhedron is called a possible faceting of the ideal crystal structure.
There are 32 well-known crystal classes, or 32 crystallographic point groups. Among them is the symmetry group of the possible faceting calculated taking account of the nodes of the ideal crystal structure belonging to it. A cyclic subgroup $C_n$ of the symmetry group of any possible faceting has order $n\le 4$ or $n=6$.
Faceting without ‘filling’. In this paper we construct two crystal structures in which there are crystal polyhedra whose symmetry groups, calculated without taking account of the nodes of the crystal structure belonging to it, have rotation axes of orders $n=8$ and $n=12$. In both cases, the crystal polyhedron is a right prism of finite height. Without taking account of the internal structure, a possible faceting of a crystal structure in three-dimensional Euclidean space cannot have an axes of rotation of order $n$ satisfying $6The proposed constructions are accompanied by a detailed analysis of ideal crystal structures, as well as Delone sets $S$ of type $(r, R)$ in $\mathbb R^2$ and $\mathbb R^3$. In particular, we produce an expanded proof of one of the theorems stated in 2010 at an international conference dedicated to the 120th anniversary of B. N. Delone.

Көрсету

Жасыру

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(3):187-226

187-226

(Rus)

(JATS XML)

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