Open Diophantine Problems

Algebraic and transcendental solutions of some exponential equations

We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.

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 number…

On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations

We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers $1$, a characteristic Sturmian number, and an arbitrary Sturmian number with the same slope.

Diophantine approximation on Veech surfaces

We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.

Diophantine approximation by conjugate algebraic integers

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $ξ$ that is either transcendental or algebraic of sufficiently large degree…

Uniform Diophantine approximation and best approximation polynomials

Let $ζ$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(ζ)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of numbers introduced by Schmidt and Summerer, and transference of the original approximation problem in dimension $n$ to suitable higher dimensions. For large $n$, we can provide an unconditional bound of order $\widehat{w}_{n}(ζ)\leq 2n-2+o(1)$. While this improves the bound of order $2n-\frac{3}{2}+o(1)$ due to Bugeaud and the author, it is unfortunately slightly wea…

单位圆上代数数的论证属性

Arithmetic properties of arguments of algebraic numbers on the unit circle

研究单位圆上代数数的论证属性，证明其迪奥菲安性和超越性，扩展了关于平面曲线几何和弹道动力学的问题。

Counting real algebraic numbers with bounded derivative of minimal polynomial

In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…

Jarník集与迪奥菲内分数集合交集的Hausdorff维度

Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals

研究了Jarník集与迪奥菲内分数集合交集的Hausdorff维度，证明其为满维，并指出无理数指数大于2且部分商有界的集合维度为1。

On uniform approximation to real numbers

Let $n \ge 2$ be an integer and $ξ$ a transcendental real number. We establish several new relations between the values at $ξ$ of the exponents of Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$, and $\hat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\hat{w}_{n}(ξ) \le 2n-1$ proved by Davenport and Schmidt in 1969.

Problems concerning Diophantine exponents of lattices

In this paper we give a survey of what is currently known about Diophantine exponents of lattices and propose several problems.

On a Problem in Diophantine Approximation

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

Diophantine methods for exponential sums, and exponential sums for Diophantine problems

Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.

A question proposed by K. Mahler on exceptional sets of transcendental functions with integer coefficients: solution of a Mahler's problem

In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic in the unit ball, with integer coefficients. This solves a strong version of an old question proposed by K. Mahler (1976).

On some open problems in Diophantine approximation

We discuss several open problems in Diophantine approximation. Among them there are famous Littlewood's and Zaremba's conjectures as well as some new and not so famous problems.

A diophantine problem on biquadrates revisited

In this paper we obtain a new parametric solution of the problem of finding two triads of biquadrates with equal sums and equal products.

Diophantine Inequalities as a Problem of Difference between Consecutive Primes

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.

On a variant of Pillai's problem with transcendental numbers

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | α^n - β^m | \leq x $ when $x$ tends to infinity. Here $α, β$ are given multiplicatively independent complex numbers with $|α| > 1$ and $|β|>1$.

Minimal relations and the Diophantine Frobenius Problem in embedding dimension three

This paper provides a formula for the minimal relations and the Frobenius number of a numerical semigroup minimally generated by three pairwise coprime positive integers.

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution.

丢番图逼近中的Littlewood与Duffin–Schaeffer型问题

Littlewood and Duffin--Schaeffer-type problems in diophantine approximation

该研究在丢番图逼近领域，建立了Gallagher定理的完全非齐次版本、丢番图纤维细化结果及Liouville纤维的尖锐阈值，还证明了一类非单调逼近函数下Duffin–Schaeffer猜想的非齐次版本。

On the Diophantine system involving pairs of triangles with the same area and the same perimeter

Many authors studied the problem that rational triangle pairs (triangle-parallelogram pairs) with the same area and the same perimeter. They investigated this problem by solving the rational solutions of the corresponding Diophantine equations. In this paper, we give a unified description of this problem by using the affine transformation in a rectangular coordinate system. According to the fact that two triangles with the same area and the same perimeter determine an affine transformation, this problem can be reduced to solving a specific Diophantine system. Moreover, we will give some ration…

An estimate for Frobenius' diophantine problem in three dimensions

We give upper and lower bounds for the largest integer not representable as positive linear combination of three given integers, disproving an upper bound conjectured by Beck, Einstein and Zacks.

On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations

We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W. Schlag (2001).

A geometric approach to the diophantine Frobenius problem

It turns out that all instances of the diophantine Frobenius problem for three coprime a_i have a common geometric structure which is independent of arithmetic coincidences among the a_i. By exploiting this structure we easily obtain Johnson's formula for the largest non-representable z, as well as a formula for the number of such z. A procedure is described which computes these quantities in O(log(max a_i)) steps.

Remarks on Diophantine approximation in function fields

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

Diophantine approximation on lines with prime constraints

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.

Exponents of Diophantine approximation

We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.

Restricted simultaneous Diophantine approximation

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.

Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.

Diophantine problems related to cyclic cubic and quartic fields

We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.

The paucity problem for certain symmetric Diophantine equations

Let $φ_1,\ldots ,φ_r\in \mathbb Z[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}(φ_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\ldots <k_r=k$. Subject to the condition $k_1+\ldots +k_r\ge \tfrac{1}{2}k(k-1)+2$, we show that there is a paucity of non-diagonal solutions to the Diophantine system $φ_j(\mathbf x)=φ_j(\mathbf y)$ $(1\le j\le r)$.

正则 episturmian 词的初始非重复复杂度及其丢番图指数

Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents

本文提出评估正则 episturmian 词初始非重复复杂度的方法，发展广义 Ostrowski 记数系统理论，并以此刻画其丢番图指数的有限性与下界，进而发现无理指数严格大于2的新不可数超越数类。

Rank zero elliptic curves induced by rational Diophantine triples

Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first su…

最佳丢番图逼近与多维三距离定理

Best Diophantine Approximations and Multidimensional Three Distance Theorem

本文基于Chevallier 1996年的引理，建立最佳丢番图逼近与多维三距离定理的联系，从前者已知结果推导后者已知结果，获该问题下确界版本新结果，并探讨逆问题的应用。

丢番图指数的几何

Geometry of Diophantine exponents

本综述探讨了丢番图指数及其满足的关系，介绍了经典丢番图指数、带权与乘性类似物及格的指数，并重点关注转换原理。

Transcendental Numbers and the Lambert-Tsallis Function

To decide upon the arithmetic nature of some numbers may be a non-trivial problem. Some cases are well know, for example exp(1) and W(1), where W is the Lambert function, are transcendental numbers. The Tsallis q-exponential, e_q (z), and the Lambert-Tsallis W_q (z) function, where q is a real parameter, are, respectively, generalizations of the exponential and Lambert functions. In the present work we use the Gelfond-Schneider theorem in order to show the arithmetic conditions on q and z such that W_q (z) and exp_q (z) are transcendental.

关于与哈代-拉马努金问题相关的斐波那契和卢卡斯序列的丢番图方程

On Diophantine equations with Fibonacci and Lucas sequences associated to Hardy-Ramanujan problem

该研究围绕哈代-拉马努金问题，探究涉及斐波那契序列和卢卡斯序列的两类三次丢番图方程，找出了兼具哈代-拉马努金数属性的斐波那契数与卢卡斯数。

Diophantine Quadruples and Cayley's Hyperdeterminant

A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Such Diophantine sets have been used to construct high rank elliptic curves. Here I demonstrate a link with Cayley's hyperdeterminants which provides a fruitful generalisation of Diophantine quadruples.

A note on zero-one laws in metrical Diophantine approximation

In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems.

A diophantine problem concerning third order matrices

In this paper we find a third order unimodular matrix, none of whose entries is $1$ or $-1$, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square integer matrices $(a_{ij})$, none of the integers $a_{ij}$ being $1$ or $-1$, such that $\det{(a_{ij})}=k$ and $\det{(a_{ij}^3)}=k^3$, where $k$ is a nonzero integer.

On a mixed Khintchine problem in Diophantine approximation

We establish a `mixed' version of a fundamental theorem of Khintchine within the field of simultaneous Diophantine approximation. Via the notion of ubiquity we are able to make significant progress towards the completion of the metric theory associated with mixed problems in this setting. This includes finding a natural mixed analogue of the classical Jarník-Besicovich Theorem. Previous knowledge surrounding mixed problems was almost entirely restricted to the multiplicative setup of de Mathan & Teulié [21], where the concept originated.

Some diophantine problems concerning a pair of rational triangles with a common circumradius

A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a common inradius or a common area. While several similar problems concerning pairs of rational triangles have been considered in the existing literature, these three problems have not been studied till now. For each of these problems, we give a parametric solution and we also indicate how additional parametric solutions of these problems may be obtained.

乘法矩阵方程的可判定性及相关丢番图问题

Decidability of multiplicative matrix equations and related Diophantine problems

本文研究了代数数域上乘法矩阵方程的可判定性，通过有效方法分析丢番图问题，提出了解的大小和数量的显式界，并探讨了对称矩阵系统的相关计数问题。

On a problem in simultaneous Diophantine approximation: Schmidt's conjecture

For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.

On a Diophantine problem with two primes and s powers of two

We refine a recent result of Parsell on the values of the form $λ_1p_1 + λ_2p_2 + μ_1 2^{m_1} + ...m + μ_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $λ_1 / λ_2$ is negative and irrational and $λ_1 / μ_1$, $λ_2/μ_2 \in \Q$.

A Diophantine problem with prime variables

We study the distribution of the values of the form $λ_1 p_1 + λ_2 p_2 + λ_3 p_3^k$, where $λ_1$, $λ_2$ and $λ_3$ are non-zero real number not all of the same sign, with $λ_1 / λ_2$ irrational, and $p_1$, $p_2$ and $p_3$ are prime numbers. We prove that, when $1 < k < 4 / 3$, these value approximate rather closely any prescribed real number.

A diophantine system

In this paper we find a parametric solution to the hitherto unsolved problem of finding three positive integers such that their sum, the sum of their squares and the sum of their cubes are simultaneously perfect squares.

Exponents of inhomogeneous Diophantine Approximation

In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.

Best Simultaneous Diophantine Approximations under a Constraint on the Denominator

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors.

On the Diophantine equations f (x) = g(y)

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for which the Diophantine equatio (y + q1 )(y + q2 ) .... (y + qm ) = f(x) has finitely many solutions in integers. Also assuming ABC Conjecture, we study the conditions for finiteness of integer solutions of the Diophantine equation f(x) = g(y).

Diophantine approximation in small degree

We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.

On the size of Diophantine $m$-tuples in imaginary quadratic number rings

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that $m\leqslant 42$. Our proof is relatively simple compared to the proofs of the similar results in positive integers.

A Survey on the Ternary Purely Exponential Diophantine Equation $a^x + b^y = c^z$

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.