Diophantine Approximation Research Articles

Diophantine approximation on the parabola with non-monotonic approximation functions

AbstractWe show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

Remarks on Diophantine approximation in function fields

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

One-sided Diophantine approximations

The paper deals with best one-sided (lower or upper) Diophantine approximations of the th kind (). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction to be a best lower or upper Diophantine approximation of the th kind to a given . The sets of best lower and upper approximations are examined in terms of their cardinalities and metric properties. Applying our results in spectral analysis, we obtain an explanation for the rarity of so-called Bethe–Sommerfeld quantum graphs.

To the reader

Dear Reader: The Moscow Journal of Combinatorics and Number Theory was founded in 2010 by the Moscow Institute of Physics and Technology, and in 2018 it started being published by MSP (Mathematical Sciences Publishers), a nonprofit scientific publisher based in Berkeley, California. Our journal publishes original, high-quality research articles from a broad range of interests within combinatorics, number theory and allied areas. Since 2011 we have published over 100 papers. Among our authors are such mathematicians as Noga Alon, Antal Balog, Jean-Pierre Demailly, Dominic Foata, Peter Frankl, Aleksandar Ivic, Sergei Konyagin, Yuri Nesterenko, Janos Pach, Yakov Sinai, Andrzej Schinzel, Wolfgang Schmidt, Carlo Viola, Michel Waldschmidt, and many others. This issue 1 of volume 8 is the first issue to appear under MSP's aegis. It contains selected papers presented by the participants of the Vilnius Conference in Combinatorics and Number Theory, which was organized with support from our journal and took place at the Department of Mathematics and Informatics of the University of Vilnius, Lithuania, 16–22 July 2017. Previous conferences connected to our journal were held in Russia (Diophantine analysis, Astrakhan, 30 July to 3 August 2012), Lithuania (Palanga Conference in Combinatorics and Number Theory, 1–7 September 2013), again Russia (Moscow Workshop in Combinatorics and Number Theory, January 27 to 2 February 2014), and Denmark (Diophantine Approximation and Related Topics, Aarhus, 13–17 July 2015). A collection of papers from the Astrakhan conference appeared in issue 3–4 of volume 3 (2013), and papers related to the Aarhus conference in issue 2–3 of volume 6 (2016). We hope that you will enjoy this issue and support the journal both with your submissions and by recommending a subscription to your institutional library!

On irrationality measure arctg 1/2

An evaluation of irrationality measure for various transcendental numbers is one of the field in diophantine approximation theory. Starting with the works of Е. Borel at the end of 19th century, were developed both general methods of evaluation for classes of some functions values and specialized approaches for estimating peculiar numbers. Diverse methods particularly were practiced for the investigating of arithmetic properties of the function $$\arctg x$$ values. For getting evaluation on irrationality measure of $$\arctg x $$ values many authors regarded them as particular case of Gauss hypergeometric function. One of the first such kind of papers was the article of M. Huttner 1987, who proved a generalized theorem about estimation on irrationality measure of the Gauss hypergeometric function values $$F_2^1\left(1,\frac{1}{k},1+\frac{1}{k}|\varepsilon x^k\right), k\in\mathbb N, k\geq 2, \varepsilon=\pm 1.$$ A big role in progress of theme have been played by works of A. Heimonen, T. Matala=aho, K. V\{a}\{a}n\{a}nen, in which was also constructed a method for evaluation on irrationality measure of the Gauss hypergeometric function values of the form $$F_2^1\left(1,\frac{1}{2},1+\frac{1}{2}|z\right), k\in \mathbb N, k\geq 2,$$ including $$F_2^1\left(1,\frac{1}{2},\frac{3}{2}|-z^2\right)=\frac {1}{z}\arctg z.$$ The approach considered by them had used approximation of the Gauss hypergeometric function by Jacobi type polynomials and gave a lot of concrete results. Last decades for evaluation of various numbers were broadly spreading methods, which used symmetric on some changes of variable integrals. Originally, integral qualitatively using the property of symmetry was applied by V.Kh.Salikhov, who used it to got the new estimate for $$\ln 3.$$ A little later V. Kh. Salikhov had applied similar symmetrized complex integral for obtaining new evaluation of $$\pi.$$ In that work he put to use classical equality $$\frac{\pi}{4}=\arctg \frac{1}{2}+\arctg \frac{1}{3}.$$ The same method, i.e. complex symmetrized integral was used by E. B. Tomashevskaya, who had estimated values of $$\arctg \frac{1}{n}, n\in\mathbb N, n>2$$ and some of previous results for such numbers were improved by her. Later on E. B. Tomashevskaya had elaborated analogical integral for estimation of $$\arctg\frac{1}{2},$$ which one had allowed to prove the best result until now $$\mu(\arctg \frac{1}{2})\leq 11.7116....$$ In 2014 K. Wu and L. Vang improved the result of V. Kh. Salikhov for $$\ln 3,$$ applying a new type integral construction, which also had used a property of symmetry. In present paper we took the idea of K. Wu and L. Vang and applied it to the integral of E. B. Tomashevskaya. It allowed us to improve arithmetic properties of integral and obtain better result for extent of irrationality $$\arctg\frac{1}{2}.$$

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.

Diophantine approximation of the orbits in topological dynamical systems

We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let $ (X, d) $ be a compact metric space and $ T:X\to X $ a continuous transformation on $ X $. For any integer valued sequence $ \{a_n\} $ and $ y\in X $, define\begin{document}$ E_y(\{a_n\}) = \bigcap\limits_{\delta&gt;0}\Big\{x\in X: T^nx\in B_{a_n}(y, \delta), \ {\text{for infinitely often}}\ n\in \mathbb N\Big\}, $\end{document}the set of points whose orbit can well approximate a given point infinitely often, where $ B_n(x, r) $ denotes the Bowen-ball. It is shown that\begin{document}$ h_{\text {top}}(E_y(\{a_n\}), T) = \frac{1}{1+a}h_{\text {top}}(X, T), \ \ {\text{with}}\ a = \liminf\limits_{n\to\infty}\frac{a_n}{n}, $\end{document}if the system $ (X, T) $ has the specification property. Here $ h_{\text {top}} $ denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.

Trigonometric sums in the metric theory of Diophantine approximation

Trigonometric sums in the metric theory of Diophantine approximation

Approximation diophantienne et distribution locale sur une surface torique

In this article we study Diophantine approximation and local distribution of a rational point on a toric surface obtained as a blow-up of $\mathbb{P}^1\times\mathbb{P}^1$. It turns out that outside a Zariski closed subset the best approximations are achieved through a family of nodal curves. Hence the investigation is reduced to the question of local distribution of a quadratic point on the projective line.

Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.

THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS

We consider sequences of the form $\left(a_{n} \alpha\right)_{n}$ mod 1, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\alpha$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.

Об оценке меры иррациональности чисел вида $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}}$ и $\frac{1}{\sqrt{k}}\arctg{\frac{1}{\sqrt{k}}}^1$

Арифметические свойства значений гипергеометрической функции изучались различными методами, начиная с работы К.~Зигеля 1929 г.. Это направление теории диофантовых приближений исследовалось такими авторами как М.~Хата [1]-[2], Ф.~Аморозо и К.~Виола [3], А.~Хеймонен, В.~Матала-Ахо и К.~Ваананен [4]-[5] и многими другими. В последние десятилетия был получен ряд интересных результатов в этой области, усилено много ранее известных оценок меры иррациональности, как для значений гипергеометрической функции, так и для других величин.В настоящее время одним из широко применяемых подходов при построении оценок показателя иррациональности является использование интегральных конструкций, симметричных относительно какой-либо замены параметров. Симметризованные интегралы и ранее использовались разными авторами, например, в работе Дж.~Рина [6], но наиболее активное развитие это направление приобрело после работы В.~Х.~Салихова [7], получившего с помощью симметризованного интеграла новую оценку для $\ln{3}$. Впоследствии симметричность различного типа позволила доказать ряд значимых результатов. Были получены новые оценки для некоторых значений логарифмической функции, функции $\arctg{x}$, классических констант (см., например, [8] -- [18]). В 2014~г., используя общие симметризованные многочлены первой степени вида $At-B$, где $t=(x-d)^2$, К.~Ву и Л.~Ванг усилили результат В.~Х.~Салихова о мере иррациональности $\ln{3}$ (см.[19]). В работе [20] идея симметричности была применена к интегралу Р.~Марковеккио, доказавшего ранее новую оценку для $\ln{2}$ в [21], что позволило улучшить результат для $\pi/3$.Данная статья является продолжением работы [22], обобщающей результаты для двух типов симметричных интегральных конструкций. Первая позволяет более эффективно оценить показатели иррациональности чисел вида $\sqrt{d}\ln{\frac{\sqrt{d}+1}{\sqrt{d}-1}}$ при $d=2^{2k+1}, d=4k+1$ для некоторых $k\in\mathbb N$ (см. [22]). Используя данный интеграл, также можно получить оценки меры иррациональности чисел $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}},\ k\in\mathbb N$. Вторая рассматриваемая интегральная конструкция дает возможность оценивать меру иррациональности некоторых значений логарифмической функции, используя симметричность другого типа, что было подробно рассмотрено в [22]. Данный интеграл позволяет также оценивать меру иррациональности значений $\frac{1}{\sqrt{k}}\arctg{\frac{1}{\sqrt{k}}}$. Обобщение этого случая предлагается в данной работе.

Explicit Salem sets and applications to metrical Diophantine approximation

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$ E(Q,\Psi,\theta) = \{ x \in \mathbb{R} : \| q x - \theta \| \leq \Psi(q) \text{ for infinitely many $q \in Q$} \}. $$ We prove a lower bound on the Fourier dimension of $E(Q,\Psi,\theta)$. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of $E(Q,\Psi,\theta)$ in new cases. We also prove a higher-dimensional analog of our result.

Симметризованные многочлены в задаче оценки меры иррациональности числа ln31

An estimate of the irrationality measure of various transcendental numbers is one of the directions in the theory of Diophantine approximations foundations. Nowadays there is a range of methods which make possible to obtain similar estimates for the values of analytic functions. The most effective method is the adding of various integral constructions; one of the first early constructions is the classical intuitive representation of the Gauss hypergeometric function. Lower estimates of the irrationality measure of rational numbers logarithms were considered by many foreign authors: А. Baker and G. Wüstholz [4], А. Heimonen, Т. Matala-aho, К. Vaänanen [5], Q. Wu [6], G. Rhin and P. Toffin [7]. In their works they used various integral constructions, giving small linear forms from logarithms and other numbers, calculated asymptotic of integrals and coefficients of the linear forms using the saddle point method, Laplace theorem, evaluated the denominator coefficients of the linear forms using various schemes "reduction of prime numbers". Review of some methods from the theory of diophantine approximation of rational numbers logarithms at that time was introduced in 2004 by V. Zudilin [8]. Then V. Kh. Salikhov in [3] considerably improved estimate of the irrationality measure of ln3, based on the same asymptotic methods, but used a new type of integral construction, which has property of summetry. Subsequently, V. Kh. Salikhov due to usage of already complex symmetrized integral improved estimate of the irrationality measures of π [15]. In the future, this method (as applied to diophantine approximation of logarithms of rational numbers) was developed by his pupils: E. S. Zolotuhina [10, 11], M. Yu. Luchin [12, 13], Е. B. Tomashevskaya [14]. It led to improvement of the irrationality measure estimates for the following numbers: µ (log(5/3)) ≤ 5.512... [14], µ (log(8/5)) &lt; 5.9897 [12], µ (log(7/5)) ≤ 4.865... [14], µ (log(9/7)) ≤6 3.6455... [10], µ (log(7/4)) &lt; 8.1004 [13]. In this paper due to usage the symmetrized real integral we obtain a new estimate of the irrationality measure of ln3. The previous irrationality measure estimate of ln3 was received in 2014 by Q. Wu and L. Wang [1]. The estimate improvement had resulted from the addition of a special square symmetrized polynomial to the symmetrized polynomials used in the integral construction of K. Wu and L. Wang.

Распределение нулей невырожденных функций на коротких отрезках

In this paper, we obtain estimates from above and from below the number of zeros of functions of a special kind, as well as an estimate of the measure of the set of points in which such functions take small values. Let f 1 ( x ), ..., f n ( x ) function defined on an interval I, n + 1 times differentiable and Wronskian of derivatives almost everywhere (in the sense of Lebesgue measure) on I different from 0. Such functions are called nondegenerate. The problem of distributing zeros of F ( x ) = a n f n ( x ) + ... + a 1 f 1 ( x ) + a 0 , a j ∈ Z , 1 ≤ j ≤ n is a generalization of many problems about the distribution of zeros of polynomials is important in the metric theory of Diophantine approximations. An interesting fact is that there is a lot in common in the distribution of roots of the function F ( x ) and the distribution of zeros of polynomials. For example, the number of zeros of F ( x ) on a fixed interval does not exceed n, as well as for polynomials — the number of zeros does not exceed the polynomial degree. Three theorems were proved: on the evaluation of the number of zeros from above, on the evaluation of the number of zeros from below, as well as an auxiliary metric theorem, which is necessary to obtain estimates from below. While obtaining lower bounds method was used for major and minor fields, who introduced V. G. Sprindzuk. Let Q &gt; 1 be a sufficiently large integer, and the interval I has the length Q − γ , 0 ≤ γ &lt; 1. Produced estimates on the top and bottom for the number of zeros of the function F ( x ) on the interval I, with | a j | ≤ Q, 0 ≤ γ &lt; 1, and also indicate the dependence of this quantity from the interval I. When γ = 0 similar results are available from A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevich, N. V. Budarina.

Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S-unit arguments, generalizing to an arbitrary number of variables results of Bugeaud–Corvaja–Zannier, Hernandez–Luca, and Corvaja–Zannier. In closely related results, and in line with observations of Silverman, we prove special cases of Vojta’s conjecture for blowups of toric varieties. As an application, we classify when terms from simple linear recurrence sequences can have a large greatest common divisor (in an appropriate sense). The primary tool used in the proofs is Schmidt’s Subspace Theorem from Diophantine approximation.

The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

Markov’s theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov’s theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the problems: How far can a straight line crossing a triangle stay away from the vertices? How far can it stay away from all vertices of the tessellation generated by this triangle? Definite binary quadratic forms are briey discussed in the last section.

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.

A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation – CORRIGENDUM

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Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ1, λ2,..., λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any e > 0; we consider the inequality |λ1p1 + λ2p 2 2 + λ3p 3 3 + λ4p 4 4 + λ5p 5 k + η | < (max pj)-σ(k)+e has infinitely many solutions in prime variables p1, p2,...,p5, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.