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Victor Beresnevich

Publications and source records attributed to Victor Beresnevich.

At least 37 records · Page 2Linked to original sources
arXiv2020

Quantitative non-divergence and Diophantine approximation on manifolds

The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.

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arXiv2018

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law for geodesics.

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arXiv2017

Sums of reciprocals and the three distance theorem

In this paper we investigate the sums of reciprocals to an arithmetic progression taken modulo one, that is sums of $\{nα-γ\}^{-1}$, where $α$ and $γ$ are real parameters and $\{\,\cdot\,\}$ is the fractional part of a real number. Bounds for these sums have been studied for a long while in connection with various applications. In this paper we develop an alternative technique for obtaining upper bounds for the sums and obtain new efficient and fully explicit results. The technique uses the so-called three distance theorem.

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arXiv2017

Diophantine approximation on manifolds and lower bounds for Hausdorff dimension

Given $n\in\mathbb{N}$ and $τ>\frac1n$, let $\mathcal{S}_n(τ)$ denote the classical set of $τ$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-τ-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ>\frac1n$ almost all points on $\mathcal{M}$ are not $τ$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set $\mathcal{M}\cap\mathcal{S}_n(τ)$. In this paper we suggest a new approach based on the Mass Transference Principle, which enables us to find a sharp lower bound for $\dim \mathcal{M}\cap\mathcal{S}_n(τ)$ for any $C^2$ submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ$ satisfying $\frac1n\leτ<\frac1m$. Here $m$ is the codimension of $\mathcal{M}$. We also show that the condition on $τ$ is best possible and extend the result to general approximating functions.

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arXiv2017

Inhomogeneous dual Diophantine approximation on affine subspaces

We prove the convergence and divergence cases of an inhomogeneous Khintchine-Groshev type theorem for dual approximation restricted to affine subspaces in $\mathbb{R} ^n$. The divergence results are proved in the more general context of Hausdorff measures.

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arXiv2017

Systems of small linear forms and Diophantine approximation on manifolds

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprindžuk from the 1970s on approximations to several real numbers by conjugate algebraic numbers. Our main result is a Khintchine type theorem, which convergence case is established without usual monotonicity constrains and the divergence case is proved for Hausdorff measures. The result encompasses several previous findings and, within the setup considered, gives the best possible improvement of a recent theorem of Aka, Breuillard, Rosenzweig and Saxcé on extremality.

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arXiv2017

A Mass Transference Principle for systems of linear forms and its applications

In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].

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arXiv2017

Sums of reciprocals of fractional parts and multiplicative Diophantine approximation

There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(α,γ):=\sum_{n=1}^N\frac{1}{n\|nα-γ\|}~~~\text{and}~~~ R_N(α,γ):=\sum_{n=1}^N\frac{1}{\|nα-γ\|}\,, \] where $α$ and $γ$ are real parameters and $\|\cdot\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \[ \{1\le n\le N:\|nα-γ\|<\varepsilon\} \, , \] valid for all sufficiently large $N$ and all sufficiently small $\varepsilon$. This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in $\mathbb{R}^2$. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.

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arXiv2016

Badly approximable points on manifolds

This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of $R^n$ has full dimension. One of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any degree.

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arXiv2016

Metric Diophantine Approximation: aspects of recent work

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'ık, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

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arXiv2015

Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.

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arXiv2015

Diophantine Approximation and applications in Interference Alignment

This paper is motivated by recent applications of Diophantine approximation in electronics, in particular, in the rapidly developing area of Interference Alignment. Some remarkable advances in this area give substantial credit to the fundamental Khintchine-Groshev Theorem and, in particular, to its far reaching generalisation for submanifolds of a Euclidean space. With a view towards the aforementioned applications, here we introduce and prove quantitative explicit generalisations of the Khintchine-Groshev Theorem for non-degenerate submanifolds of $\mathbb{R}^n$. The importance of such quantitative statements is explicitly discussed in Section 4.7.1 of Jafar's monograph `Interference Alignment - A New Look at Signal Dimensions in a Communication Network', Foundations and Trends in Communications and Information Theory, Vol. 7, no. 1, 2010.

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arXiv2015

Integral polynomials with small discriminants and resultants

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$: (i) given $0\le v\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\in\mathcal{P}_n(Q)$ such that $$0<|D(P)|\le Q^{2n-2-2v};$$ (ii) given $0\le w\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P_2\in\mathcal{P}_n(Q)$ such that $$0<|R(P_1,P_2)|\le Q^{2n-2w}.$$ Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R.C.Vaughan and S.Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

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arXiv2014

Badly approximable points on planar curves and winning

For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \in R^2 such that max \{ ||qx||^{1/i}, \, ||qy||^{1/j} \} > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) \cap C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we obtain the inhomogeneous generalizations of the winning results for planar curves and lines and also show that the inhomogeneous form of Bad(i,j) is winning for two dimensional Schmidt games.

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arXiv2013

Non-planarity and metric Diophantine approximation for systems of linear forms

In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is introduced and studied. This notion generalises the one of non-planarity in $\Bbb R^n$ and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

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arXiv2013

Multiplicative zero-one laws and metric number theory

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin-Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.

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arXiv2013

A note on three problems in metric Diophantine approximation

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider three inhomogeneous problems that further develop these classical results. Firstly, we obtain a Jarnik type theorem for the set of multiplicatively approximable points in the plane. This Hausdorff measure statement does not reduce to Gallagher's Lebesgue measure statement as one might expect and is new even in the homogeneous setting. Next, we establish a Jarnik type theorem for the set of multiplicatively approximable points on a non-degenerate planar curve. This completes the Hausdorff theory for planar curves. Finally, we show that the set of simultaneously inhomogeneously (i,j)-badly approximable points in the plane is of full dimension. The underlying philosophy behind the proof has other applications; e.g. towards establishing the inhomogeneous version of Schmidt's Conjecture. The higher dimensional analogues of the planar results are also discussed.

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