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Dmitry Gayfulin

Publications and source records attributed to Dmitry Gayfulin.

14 recordsLinked to original sources
arXiv2026

Sums of Laurent series with bounded partial quotients

In 1947 M.Hall proved that every real number is the sum of an integer and two real numbers whose partial quotients are at most $4$. Later, Cusick proved that every real number is the sum of an integer and two real numbers whose partial quotients are at least $2$. In a recent paper, the authors proved that every real number is the sum of two real numbers whose partial quotients diverge. In this paper, we prove an analogue of these results for Laurent series.

math.NT↗
arXiv2026

A note on exact approximations

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

math.NT↗
arXiv2024

Hausdorff dimension estimates for Sudler products with positive lower bound

Given an irrational number $α$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(α) = \prod_{r=1}^N 2\lvert \sin πr α\rvert$. We show that $\liminf_{N \to \infty} P_N(α) >0$ and $\limsup_{N \to \infty} P_N(α)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds 3 only finitely often, which confirms a conjecture of the second-named author and partially answers a question of J. Shallit. Furthermore, we show that the Hausdorff dimension of the set of those $α$ that satisfy $\limsup_{N \to \infty} P_N(α)/N < \infty,\liminf_{N \to \infty} P_N(α) >0$ lies between $0.7056$ and $0.8677$, which makes significant progress in a question raised by Aistleitner, Technau, and Zafeiropoulos. We also show that the set of such $α$ is invariant under the Gauss map $T$.

math.NT↗
arXiv2023

Additive Model Perturbations Scaled by Physical Tendencies for Use in Ensemble Prediction

Imperfections and uncertainties in forecast models are often represented in ensemble prediction systems by stochastic perturbations of model equations. In this article, we present a new technique to generate model perturbations. The technique is termed Additive Model-uncertainty perturbations scaled by Physical Tendencies (AMPT). The generated perturbations are independent between different model variables and scaled by the local-area-averaged modulus of physical tendency. The previously developed Stochastic Pattern Generator is used to generate space and time-correlated pseudo-random fields. AMPT attempts to address some weak points of the popular model perturbation scheme known as Stochastically Perturbed Parametrization Tendencies (SPPT). Specifically, AMPT can produce non-zero perturbations even at grid points where the physical tendency is zero and avoids perfect correlations in the perturbation fields in the vertical and between different variables. Due to a non-local link from physical tendency to the local perturbation magnitude, AMPT can generate significantly greater perturbations than SPPT without causing instabilities. Relationships between biases and spreads caused by AMPT and SPPT were studied in an ensemble of forecasts. The non-hydrostatic, convection-permitting forecast model COSMO was used. In ensemble prediction experiments, AMPT perturbations led to statistically significant improvements (compared to SPPT) in probabilistic performance scores such as spread-skill relationship, CRPS, Brier Score, and ROC area for near-surface temperature. AMPT had similar but weaker effects on near-surface wind speed and mixed effects on precipitation.

physics.ao-ph↗
arXiv2023

Discrete part of the second Lagrange spectrum

Given an irrational number $α$ consider its irrationality measure function $ψ_α(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z}}\|qα\|$. The set of all values of $λ(α)=(\limsup\limits_{t\to\infty} tψ_α(t))^{-1}$ where $α$ runs through the set $\mathbb{R}\setminus\mathbb{Q}$ is called the Lagrange spectrum $\mathbb{L}$. In a paper by Moshchevitin an irrationality measure function $ψ^{[2]}_α(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z},q\ne q_i}\|qα\|$ was introduced. In other words, we consider the best approximations by fractions, whose denominators are not the denominators of the convergents to $α$. Replacing the function $ψ_α$ in the definition of $\mathbb{L}$ by $ψ^{[2]}_α$, one can get a set $\mathbb{L}_2$ which is called the ''second'' Lagrange spectrum. In this paper we give the complete structure of discrete part of $\mathbb{L}_2$.

math.NT↗
arXiv2023

On Furstenberg's Diophantine result

We give a very simple and explicit exposition of the effective results on $\times a\times b$ by Bourgain, Lindenstrauss, Michel and Venkatesh.

math.NT↗
arXiv2021

On the derivative of the Minkowski question-mark function

The Minkowski question-mark function $?(x)$ is a continuous strictly increasing function defined on $[0,1]$ interval. It is well known fact that the derivative of this function, if exists, can take only two values: $0$ and $+\infty$. It is also known that the value of the derivative $?'(x)$ at the point $x=[0;a_1,a_2,\ldots,a_t,\ldots]$ is connected with the limit behavior of the arithmetic mean $(a_1+a_2+\ldots+a_t)/t$. Particularly, N. Moshchevitin and A. Dushistova showed that if $a_1+a_2+\ldots+a_t<κ_1 t$, where $κ_1 = 2\log\bigl({\frac{1+\sqrt{5}}{2}}\bigr)/\log{2}= 1.3884\ldots$, then $?'(x)=+\infty$. They also proved that the constant $κ_1$ is non-improvable. We consider a dual problem: how small can be the quantity $a_1+a_2+\ldots+a_t-κ_1 t$ if $?'(x)=0$? We obtain the non-improvable estimates of this quantity.

math.NT↗
arXiv2019

Diophantine properties of fixed points of Minkowski question mark function

We consider irrational fixed points of the Minkowski question mark function $? (x)$, that is irrational solutions of the equation $? (x)=x$. It is easy to see that there exist at least two such points. Although it is not known if there are other fixed points, we prove that the smallest and the greatest fixed points have irrationality measure exponent equal to 2. We give more precise results about the approximation properties of these fixed points. Moreover, in Appendix we introduce a condition from which it follows that there are only two irrational fixed points.

math.NT↗
arXiv2017

A spatio-temporal stochastic pattern generator for simulation of uncertainties in geophysical ensemble prediction and ensemble data assimilation

A generator of spatio-temporal pseudo-random Gaussian fields that satisfy the "proportionality of scales" property (Tsyroulnikov, 2001) is presented. The generator is based on a third-order in time stochastic differential equation with a pseudo-differential spatial operator defined on a limited area 2D or 3D domain in the Cartesian coordinate system. The generated pseudo-random fields are homogeneous and isotropic in spacetime (with the scaled vertical and temporal coordinates). The correlation functions in any spatio-temporal direction belong to the Matérn class. The spatio-temporal correlations are non-separable. A spectral-space numerical solver is implemented and accelerated exploiting properties of real-world geophysical fields, in particular, smoothness of their spatial spectra. The generator is designed to create additive or multiplicative, or other spatio-temporal perturbations that represent uncertainties in numerical prediction models in geophysics. The program code of the generator is publicly available.

physics.data-an↗
arXiv2017

Admissible endpoints of gaps in the Lagrange spectrum

We call a positive real number $λ$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $α$ such that $μ(α)=λ$. Here $μ(α)$ denotes the Lagrange constant of $α$ - maximal real number $c$ such that $\forall \varepsilon>0$ the inequality $|α-\frac{p}{q}|\le\frac{1}{(c-\varepsilon)q^2}$ has infinitely many solutions for relatively prime $p$ and $q$. In this paper we establish a necessary and sufficient condition of admissibility of the Lagrange spectrum element and construct an infinite series of not admissible numbers.

math.NT↗
arXiv2013

On the derivative of two functions from Denjoy-Tichy-Uitz family

The family of functions, we investigate in this article, was originally introduced by A.Denjoy and later rediscovered by R Tichy and J. Uitz. We denote the functions of the family by $g_λ(x),$ where $λ\in(0,1)$. The definition will be given in the following section. The most famous function of the family is the Minkiowski question-mark function. As we would see, it corresponds to $λ=\frac12$. All functions of the family are continuous, strictly increasing and map the segment $[0,1]$ onto itself. Moreover, they are singular i.e. $\forall λ$ the derivative $g'_λ(x),$ if exists, can take only two values: 0 and $+\infty.$ In this paper we consider two functions of the class which correspond to $λ$ equals $\frac{\sqrt5-1}2$ or $1-\frac{\sqrt5-1}2.$ The aim of this paper is to prove some theorems about essential conditions on x such that if the condition holds then the derivative $g'_λ(x)$ exists and has determined value. The constants used in our theorems are non-improvable. Our paper is wirtten in Russian. However Introduction and the formulation of main results are written in English.

math.NT↗
arXiv2013

On Diophantine exponents in dimension 4

We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.

math.NT↗