Khintchine Theorem Research Articles

An improved convergence case for Diophantine approximations on IFS fractals

The objective of this paper is to (partially) address the issue of finding an analogue to Khintchine's theorem for IFS Fractals. We study the convergence case for Diophantine approximations, and show an improved result for higher dimensions. This matter has been previously studied by Pollington and Velani in arXiv:math/0401149. Pollington and Velani show a similar result to the one in this paper (a Khinchine convergence case) and we shall show how our result is an improvement in the higher dimensional cases.

Phase-space quantum Wiener-Khintchine theorem.

We derive a quantum version of the classical-optics Wiener-Khintchine theorem within the framework of detection of phase-space displacements with a suitably designed quantum ruler. A phase-space-based quantum mutual coherence function is introduced that includes the contribution of the detector. We obtain an universal equality linking resolution with coherence. This is illustrated with the case of Gaussian states and number states.

Incremental nonlinear stability analysis of stochastic systems perturbed by Lévy noise

AbstractWe present a theoretical framework for characterizing incremental stability of nonlinear stochastic systems perturbed by either compound Poisson shot noise or finite‐measure Lévy noise. For each noise type, we compare trajectories of the perturbed system with distinct noise sample paths against trajectories of the nominal, unperturbed system. We show that for a finite number of jumps arising from the noise process, the mean‐squared error between the trajectories exponentially converge toward a bounded error ball across a finite interval of time under practical boundedness assumptions. The convergence rate for shot noise systems is the same as the exponentially stable nominal system, but with a tradeoff between the parameters of the shot noise process and the size of the error ball. The convergence rate and the error ball for the Lévy noise system are shown to be nearly direct sums of the respective quantities for the shot and white noise systems separately, a result which is analogous to the Lévy–Khintchine theorem. We demonstrate both empirical and analytical computation of the error ball using several numerical examples, and illustrate how varying the parameters of the system affect the tightness of the bound.

Non-homogeneous analysis of rogue wave probability evolution over a shoal

Non-equilibrium evolution of wave fields, as occurring over sudden bathymetry variations, can produce rogue seas with anomalous wave statistics. We handle this process by modifying the Rayleigh distribution through the energetics of second-order theory and a non-homogeneous reformulation of the Khintchine theorem. The resulting probability model reproduces the enhanced tail of the probability distribution of unidirectional wave tank experiments. It also describes why the peak of rogue wave probability appears atop the shoal, and explains the influence of depth on variations in peak intensity. Furthermore, we interpret rogue wave likelihoods in finite depth through the $H$ – $\sigma$ diagram, allowing a quick prediction for the effects of rapid depth change apart from the probability distribution.

Signal estimation and coherence.

We address the relation between quantum metrological resolution and quantum coherence. We develop a quantum version of the Wiener-Khintchine theorem for a suitable model of a quantum ruler, putting the contributions of the probe and measurement on equal footing. We illustrate this approach with several examples of linear metrology.

Khintchine's theorem for affine hyperplanes in positive charateristic

In this paper we establish the convergence case of Khintchine's theorem for affine hyperplanes in function field of positive characteristic. Along with that, we also prove a quantitative version of the same. The main technique used in the proof is a dynamical result called ‘Quantitative nondivergence’ due to D.Y. Kleinbock and G.A. Margulis ([23]).

On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function ψ:N→(0,1/2) with ∑qψ(q)=∞ and any number γ, the following setW(ψ,γ)={x∈[0,1]:|qx−p−γ|<ψ(q) for infinitely many q,p∈N} has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if γ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound conditionψ(q)=O((q(log⁡log⁡q)2)−1). In particular, this covers the case when γ is not Liouville, for example π,e,ln⁡2,2. In general, if γ is irrational, ψ(q)=O(q−1(log⁡log⁡q)−2) and in addition,(liminfQ→∞∑q=QQ(log⁡Q)1/8ψ(q))=∞, then W(ψ,γ) has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.

Densidad espectral en guías de onda mediante propiedades estadísticas y teorema de Wiener-Khintchine

There are several cases in which the use of optical fiber is use, its properties above many mechanical cables makes it an excellent waveguide, it has led to the development of communications, engineering, including the medical and biological science. However, poor coupling or cuts can affect this means of transmission. When the optical path within the waveguide undergoes alterations, there are losses, overlap in the signal, errors in demodulation and decoding, limiting its transmission capacity. In this work the result in the changes that the spectral density presents in a circular waveguide are reported, when subjected to deformations in the axes (x, y), through the perspective of speckle pattern, and its statistical properties in correlation with the Wiener Khintchine Theorem. The results obtained suggest that the behavior of the spectral density is descending and that it is feasible to determine the similarity between the reference spectral density and the one that undergoes deformation for both the statistical analysis and the Wiener Khintchine Theorem.

Variants of Khintchine's theorem in metric Diophantine approximation

New results towards the Duffin-Schaeffer conjecture, which is a fundamental problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function ψ:N→R we denote by W(ψ) the set of all x∈R such that |nx−a|<ψ(n) for infinitely many a,n. Analogously, denote W′(ψ) if we additionally require a,n to be coprime. Aistleitner et al. [2] proved that W′(ψ) is of full Lebesgue measure if there exist an ε>0 such that ∑n=2∞ψ(n)φ(n)/(n(log⁡n)ε)=∞. This result seems to be the best one can expect from the method used. Assuming the extra divergence ∑n=2∞ψ(n)/(log⁡n)ε=∞ we prove that W(ψ) is of full measure. This could also be deduced from the result in [2], but we believe that our proof is of independent interest, since its method is totally different from the one in [2]. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of ψ.

KHINTCHINE'S THEOREM WITH RANDOM FRACTIONS

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schaeffer Conjecture, like the Duffin-Schaeffer Conjecture itself, implies Catlin's Conjecture (1976). It is not obvious whether the Duffin-Schaeffer Conjecture and its random version imply one another, and it is not known whether Catlin's Conjecture implies either of them. The question of whether Catlin implies Duffin-Schaeffer has been unsettled for decades.

On the Importance of Statistical Homogeneity to the Scaling of Rain

AbstractScaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener–Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the “observed” power spectrum no longer represents a “universal” scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.

Modeling of Aggregated IoT Traffic and Its Application to an IoT Cloud

As the Internet of Things (IoT) continues to gain traction in telecommunication networks, a very large number of devices are expected to be connected and used in the near future. In order to appropriately plan and dimension the network, as well as the back-end cloud systems and the resulting signaling load, traffic models are employed. These models are designed to accurately capture and predict the properties of IoT traffic in a concise manner. To achieve this, Poisson process approximations, based on the Palm–Khintchine theorem, have often been used in the past. Due to the scale (and the difference in scales in various IoT networks) of the modeled systems, the fidelity of this approximation is crucial, as, in practice, it is very challenging to accurately measure or simulate large-scale IoT deployments. The main goal of this paper is to understand the level of accuracy of the Poisson approximation model. To this end, we first survey both common IoT network properties and network scales as well as traffic types. Second, we explain and discuss the Palm–Khintiche theorem, how it is applied to the problem, and which inaccuracies can occur when using it. Based on this, we derive guidelines as to when a Poisson process can be assumed for aggregated periodic IoT traffic. Finally, we evaluate our approach in the context of an IoT cloud scaler use case.

An analogue of Khintchine's theorem for self-conformal sets

AbstractKhintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

On the Detection of Statistical Heterogeneity in Rain Measurements

AbstractThe application of the Wiener–Khintchine theorem for translating a readily measured correlation function into the variance spectrum, important for scale analyses and for scaling transformations of data, requires that the data be wide-sense homogeneous (stationary), that is, that the first and second moments of the probability distribution of the variable are the same at all times (stationarity) or at all locations (homogeneity) over the entire observed domain. This work provides a heuristic method independent of statistical models for evaluating whether a set of data in rain is wide-sense stationary (WSS). The alternative, statistical heterogeneity, requires 1) that there be no single global mean value and/or 2) that the variance of the variable changes in the domain. Here, the number of global mean values is estimated using a Bayesian inversion approach, while changes in the variance are determined using record counting techniques. An index of statistical heterogeneity (IXH) is proposed for rain such that as its value approaches zero, the more likely the data are wide-sense stationary and the more acceptable is the use of the Wiener–Khintchine theorem. Numerical experiments as well as several examples in real rain demonstrate the potential of IXH to identify statistical homogeneity, heterogeneity, and statistical mixtures. In particular, the examples demonstrate that visual inspections of data alone are insufficient for determining whether they are wide-sense stationary. Furthermore, in this small data collection, statistical heterogeneity was associated with convective rain, while statistical homogeneity appeared in more stratiform or mixed rain events. These tentative associations, however, need further substantiation.

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

Multiple recurrence theorems for set-valued maps

In this paper, we mainly extend the Khintchine's recurrence theorem, Furstenberg's multiple recurrence theorem, and multiple Birkhoff recurrence theorem from single-valued maps to set-valued maps.

An Inhomogeneous Jarník type theorem for planar curves

AbstractIn metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental to each of them. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximation on manifolds for each of the classes above. In particular, both Khintchine and Jarník-type results have been established for approximation on planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves in the setting of dual approximation and in so doing complete the metric theory of Diophantine approximation on planar curves.

A note on badly approximabe sets in projective space

Recently, Ghosh \& Haynes \cite{HG} proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarn\'{\i}k-type result also holds for `badly approximable' points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine's theorem for convergence relating to `intrinsic' approximation of points in these compact sets.

Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations

We present results on the broadband nature of the power spectrum , , for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of intermittent maps with for , where . Such maps have summable decay of correlations when , and extends to a continuous function on by the classical Wiener–Khintchine theorem. We show that is typically bounded away from zero for Hölder observables. Moreover, in the nonsummable case , we show that is defined almost everywhere with a continuous extension defined on , and is typically nonvanishing.

Modal Parameters Directly Estimated from Power Spectral Densities or Correlation Functions in Output-Only Analysis

Output-only analysis considers the system excited either by operative or by natural forces, in both cases the input loading which excites the structure is considered, at least in a limited frequency band, as a white noise. Because it is not possible to find directly the spectra, it is necessary to pass through the correlation functions so as to apply the Wiener–Khintchine theorem in order to find the power spectral densities (PSDs). In the past, the modal parameters have been essentially derived by approaches that manipulated the functions mentioned above. In this paper, the modal parameters will be directly estimated from the PSDs, in the frequency domain, and from the correlation functions in the time domain. A particular attention is devoted to the problems regarding the damping ratio overestimation. This effect due to the limited time window in the correlation function estimate is highly present in the low-frequency modes and could bring to large estimation errors. Experimental examples, carried out both on cantilever beams and on an helicopter blade, are presented. They show the problems related to the techniques using the PSDs, or the correlation functions and the possibility to overcome the overestimation of the damping factors due to the triangular window, also known as Bartlett window.