On the spectrum of Diophantine exponents of lattices
In this paper we describe the spectrum of values of weak uniform Diophantine exponents of lattices in arbitrary dimension.
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arXiv subjects
Oleg N. German
Publications and source records attributed to Oleg N. German.
On uniform Diophantine exponents of lattices
In this paper we study the spectrum of weak uniform Diophantine exponents of lattices and obtain its complete description in the two-dimensional case.
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On the weak uniform Diophantine exponent of a real number
In this paper we introduce the notion of a weak uniform Diophantine exponent of a real number and obtain the complete description of the spectrum of its values.
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Geometry of Diophantine exponents
Diophantine exponents are ones of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation which studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle.
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On triviality of uniform Diophantine exponents of lattices
In this paper we prove that uniform Diophantine exponents of lattices attain only trivial values.
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An analogue of Mahler's transference theorem for multiplicative Diophantine approximation
Khintchine's and Dyson's transference theorems can be very easily deduced from Mahler's transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler's theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler's theorem which implies the multiplicative transference theorem immediately.
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Diophantine exponents of lattices and growth of multidimensional analogues of partial quotients
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.
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Multiparametric geometry of numbers and its application to splitting transference theorems
In this paper we consider a multiparametric version of Wolfgang Schmidt and Leonard Summerer's parametric geometry of numbers. We apply this approach in two settings: the first one concerns weighted Diophantine approximation, the second one concerns Diophantine exponents of lattices. In both settings we use multiparametric approach to define intermediate exponents. Then we split the weighted version of Dyson's transference theorem and an analogue of Khintchine's transference theorem for Diophantine exponents of lattices into chains of inequalities between the intermediate exponents we define basing on the intuition provided by the parametric approach.
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On transference principle and Nesterenko's linear independence criterion
We consider the problem of simultaneous approximation of real numbers $θ_1, \ldots,θ_n$ with rationals and the dual problem of approximating zero with the values of the linear form $x_0+θ_1x_1+\ldots+θ_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Besides that, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.
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Symmetries of a two-dimensional continued fraction
In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type ones, that correspond to the multiplication by units of the corresponding number field, and so called palindromic ones. The main result of the paper is a criterion for a two-dimensional continued fraction to have a palindromic symmetry. This criterion is analogous to the criterion for the period of a quadratic irrationality to be symmetric.
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Transference theorems for Diophantine approximation with weights
In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.
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Multiplicative parametric geometry of numbers and transference theorems for lattice exponents
In this paper we adapt parametric geometry of numbers developed by Wolfgang Schmidt and Leonard Summerer to a multiplicative setting, and derive a chain of inequalities for the corresponding exponents which splits the transference inequality for Diophantine exponents of lattices in the same way Khintchine's transference inequalities for simultaneous approximation can be split.
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Problems concerning Diophantine exponents of lattices
In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.
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Linear forms of a given Diophantine type and lattice exponents
In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.
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A short proof of Minkowski's second theorem and its application to some open questions
In this paper we present a surprisingly short proof of Minkowski's second theorem. The author hopes there is no mistake in it, though the argument seems to be too plain to contain one. Also, we apply the main construction of the proof to some problems concerning the anomaly of a convex body, the density of the densest lattice packing, lattice point enumerator, and Ehrhart polynomial.
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Palindromes and periodic continued fractions
This paper is devoted to a detailed exposition of geometry of continued fractions. We pay particular interest to the case of quadratic irrationalities and use the technique described to prove a criterion for the continued fraction of a quadratic surd to have a symmetric period.
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Diophantine exponents of lattices
In this paper we define Diophantine exponents of lattices and investigate some of their properties. We prove transference inequalities and construct some examples with the help of Schmidt's subspace theorem.
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Improvement upon Mahler's transference theorem
In this paper we obtain new transference theorems improving some classical theorems which belong to Kurt Mahler. We formulate those theorems in terms of consecutive minima of pseudo-compound parallelepipeds.
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