Inhomogeneous Diophantine Approximation Research Articles

DIOPHANTINE TRANSFERENCE PRINCIPLE OVER FUNCTION FIELDS

Abstract We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3)101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

We study inhomogeneous Diophantine approximation over the completion Kv of a function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain an effective upper bound for the Hausdorff dimension of the setBadA(ϵ)={θ∈Kvm:lim inf(p,q)∈Rvm×Rvn,‖q‖→∞lim inf(p,q)∈Rvm×Rvn,‖q‖→∞‖q‖n‖Aq−θ−p‖m⩾ϵ}, of ϵ-badly approximable targets θ in Kvm given A in Mm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of Rv-grids. We characterize matrices A for which BadA(ϵ) has full Hausdorff dimension for some ϵ>0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances.

Inhomogeneous Diophantine approximation for generic homogeneous functions

This paper is a sequel to [Monatsh. Math. 194 (2021) 523–554] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers [Formula: see text] and [Formula: see text], any [Formula: see text], and any homogeneous function [Formula: see text] that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function [Formula: see text] for a generic element [Formula: see text] in the [Formula: see text]-orbit of [Formula: see text] to be (respectively, not to be) [Formula: see text]-approximable at [Formula: see text]: that is, for there to exist infinitely many (respectively, only finitely many) [Formula: see text] such that [Formula: see text] for each [Formula: see text]. In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of [Formula: see text] that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace [Formula: see text] above by any closed subgroup of [Formula: see text] that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper.

Einige Bemerkungen über inhomogene diophantische Approximationen

We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find some of them in the literature. In particular, we prove a precise version of the Kronecker approximation theorem and a related result on coprime approximation.

S-arithmetic inhomogeneous Diophantine approximation on manifolds

We prove S-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For S consisting of more than one valuation, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the S-arithmetic inhomogeneous counterpart of the Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces in conjunction with the transference principle of Beresnevich and Velani.

On the distribution of nonprimitive lattice points in the plane

AbstractWe improve on a result by Svetlana Jitomirskaya and Wencai Liu dealing with inhomogeneous Diophantine approximation in the coprime setting.

Inhomogeneous Diophantine Approximation on M0-Sets with Restricted Denominators

Abstract Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu $ with the property that $ |\widehat{\mu }(t)| \ll (\log |t|)^{-A}$ for some constant $ A> 0 $. Let $\mathcal{A}= (q_n)_{n\in{\mathbb{N}}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A>2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (1) the points of interest are restricted to $F$ and (2) the denominators of the “shifted” rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx \textrm{mod} 1)_{n\in{\mathbb{N}}} $ is uniformly distributed for $\mu $ almost all $x \in F$. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences $\mathcal{A}$ for which the prime divisors are restricted to a finite set of $k$ primes and $A> 2k$. Loosely speaking, for such sequences, our result can be viewed as a quantitative refinement of the fundamental theorem of Davenport, Erdös, and LeVeque (1963) in the theory of uniform distribution.

A Practical Diophantine Approximation for Sparse Linear Array Direction Finding

This article presents a computationally efficient, fixed complexity algorithm for estimating the angle of arrival of a radio frequency signal that impinges on a sparse linear array of sensors. The likelihood function for the bearing estimation problem is reduced to an inhomogeneous Diophantine approximation problem involving an unknown set of integer multiples of 2π. In solving for the integers, we propose a modification to the Cassels algorithm and generate an approximate linearized maximum-likelihood (ML) estimate of bearing. The number of iterations for convergence of the algorithm can be precomputed, which allows the algorithm to have fixed computational complexity. The proposed technique applies to a large class of array configurations, allowing the array configuration to be chosen to optimize estimation performance rather than to suit the algorithm. The simulation results show the performance of the proposed bearing estimation algorithm to be very close to that of ML estimation, but with a significantly reduced computational overhead.

Inhomogeneous Diophantine approximation over fields of formal power series

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $\alpha$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, \deg Q \to \infty} \| Q \| \cdot |\langle Q \alpha - \theta \rangle| \geq \eps $$ has full Hausdorff dimension.

Inhomogeneous Diophantine approximation in the coprime setting

Given n∈N and x,γ∈R, let||γ−nx||′=min⁡{|γ−nx+m|:m∈Z,gcd⁡(n,m)=1}, Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number α and almost every γ∈R,lim infn→∞n||γ−nα||′=0 and that there exists C>0, such that for all α∈R\\Q and γ∈[0,1),lim infn→∞n||γ−nα||′<C. We prove the first conjecture and disprove the second one.

Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents

We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.

A Fourier-analytic approach to inhomogeneous Diophantine approximation

In this paper, we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,\theta)$ as follows, \begin{eqnarray*} \left\{x\in [0,1]:\left |x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\text{ for infinitely many coprime pairs of numbers } m,n\right\}, \end{eqnarray*} where $\{f(n)\}_{n\in\mathbb{N}}$ and $\{\theta(n)\}_{n\in\mathbb{N}}$ are sequences of real numbers in $[0,1/2]$. We will completely determine the Hausdorff dimension of $W(f,\theta)$ in terms of $f$ and $\theta$. As a by-product, we also obtain a new sufficient condition for $W(f,\theta)$ to have full Lebesgue measure and this result is closely related to the study of \ds with extra conditions.

Modified shrinking target problem in beta dynamical systems

In the paper, we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the β-transformation Tβ:x→βx(mod1). The Hausdorff dimension of the lim sup setW(f,φ)={x∈[0,1):|Tβnx−f(x)|<φ(n)for infinitely many n∈N} is determined, where f:[0,1]→[0,1] is a Lipschitz function and φ is a positive function defined on N. The set W(f,φ) can be viewed as a dynamical analogue of the inhomogeneous Diophantine approximation in which the inhomogeneous factor is allowed to vary.

Inhomogeneous diophantine approximation with prime constraints

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.

On Diophantine transference principles

AbstractWe provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and their nondegenerate submanifolds.

Diophantine Approximation in Prescribed Degree

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the intensely studied problem of approximation by algebraic numbers (and integers) of bounded degree. We establish the answer to a question of Bugeaud concerning approximation to transcendental real numbers by quadratic irrational numbers, and thereby we refine a result of Davenport and Schmidt from 1967. We also obtain several new characterizations of Liouville numbers, and certain new insights on inhomogeneous Diophantine approximation. As an auxiliary side result, we provide an upper bound for the number of certain linear combinations of two given relatively prime integer polynomials with a linear factor. We conclude with several open problems.

The Non-homogeneous Groshev Convergence theorem for Diophantine Approximation on Manifolds

This paper is based on Khintchine theorem, Groshev theorem and measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifolds is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold in , based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and transformation of lemma and its properties and the main inhomogeneous conversion principle, we know these two types of set in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent theory of Diophantine approximation is obtained. The main result is that Lebesgue measure is inhomogeneous strongly extremal. At last we use the fact that friendly measure is strongly contracting measure to develop an inhomogeneous strong extreme measure which is restricted to matrices with dependent quantities

Solution of Cassels’ Problem on a Diophantine Constant over Function Fields

This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $\theta\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(\theta,\gamma)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle N\theta - \gamma \rangle|$. We show that for every $\theta$ there exists $\gamma$ such that $c(\theta,\gamma)\geq q^{-2}$, and find a sufficient condition on $\theta$ which forces $c(\theta,\gamma) \leq q^{-2}$ for every $\gamma$. Given $\theta$, we prove that the set $BA_{\theta}=\left\{\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(\theta,\gamma)>0\right\}$ has full Hausdorff dimension. Our methods allow us to solve the case of vectors in $\mathbb F_q\left(\left(\frac{1}{t}\right)\right)^d$ as well. Our results offer a strengthening to analogues of results for real inhomogeneous approximation.