Diophantine sets of polynomials over number fields

Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields

We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophant…

Functional composition of polynomials: indecomposability, Diophantine equations and lacunary polynomials

Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).

Intersective polynomials and Diophantine approximation, II

By applying Schmidt's lattice method, we prove results on simultaneous Diophantine approximation modulo 1 for systems of polynomials in a single prime variable provided that certain local conditions are met.

Hausdorff and packing dimension of Diophantine sets

Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a result of Jarník and Besicovitch to intermediate exponents.

Diophantine sets and Dirichlet improvability

This note pushes further the discussion about relations between Dirichlet improvable, badly approximable and singular points held in recent joint work with Beresnevich, Guan, Velani and Ramirez, by considering Diophantine sets extending the notion of badly approximability.

On the Diophantine equation $f(x)f(y)=f(z)^n$ involving Laurent polynomials

By the theory of elliptic curves, we investigate the nontrivial rational parametric solutions of the Diophantine equation $f(x)f(y)=f(z)^n$, where $n=1,2$ and $f(X)$ are some simple Laurent polynomials.

On the Diophantine equations $z^2=f(x)^2 \pm g(y)^2$ concerning Laurent polynomials

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm g(y)^2$ for some simple Laurent polynomials $f$ and $g$.

On the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ involving Laurent polynomials

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.

Diophantine approximation of polynomials over $\mathbb{F}_q[t]$ satisfying a divisibility condition

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

Diophantine equations with Euler polynomials

In this paper we determine possible decompositions of Euler polynomials $E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation $$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y),$$ with $g\in \mathbb{Q}[X]$ of degree at least $2$ and $k\geq 7$, has only finitely many integers solutions $x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.

Exceptional sets in dynamical systems and Diophantine approximation

The nature and origin of exceptional sets associated with the rotation number of circle maps, Kolmogorov-Arnol'd-Moser theory on the existence of invariant tori and the linearisation of complex diffeomorphisms are explained. The metrical properties of these exceptional sets are closely related to fundamental results in the metrical theory of Diophantine approximation. The counterpart of Diophantine approximation in hyperbolic space and a dynamical interpretation which led to the very general notion of `shrinking targets' are sketched and the recent use of flows in homogeneous spaces of lattice…

The Arithmetic of Diophantine Approximation Groups II: Mahler Arithmetic

This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real number $θ$ we associate its polynomial diophantine approximation ring: a tri-filtered subring of a nonstandard model of the ring $\mathbb{Z}[X]$. We characterize the filtration structure of the polynomial diophantine approximation ring according to the Mahler class and the Mahler type of $θ$. The arithmetic of polynomial diophantine approximation groups is intro…

Diophantine approximations, Markov and Lagrange spectra and dynamical Cantor sets

This text is a slightly expanded version of a survey article on certain aspects of low dimensional dynamics and number theory written after a kind invitation by the editors of the Notices of the American Mathematical Society.

An effective Pila-Wilkie theorem for sets definable using Pfaffian functions, with some diophantine applications

We prove an effective version of the Pila-Wilkie Theorem for sets definable using Pfaffian functions, providing effective estimates for the number of algebraic points of bounded height and degree lying on such sets. We also prove effective versions of extensions of this result due to Pila and Habegger-Pila . In order to prove these counting results, we obtain an effective version of Yomdin-Gromov parameterization for sets defined using restricted Pfaffian functions. Furthermore, for sets defined in the restricted setting, as well as for unrestricted sub-Pfaffian sets, our effective estimates d…

Diophantine approximation on polynomial curves

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrized by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with $P(X)\equiv 1$ over the rational number field, we establish proper generalizations of their main result.

Diophantine sets in general are Cantor sets

Let $γ\in(0;\frac{1}{2}),τ\geq 1$ and define the "$γ,τ$ Diophantine set" as: $$D_{γ,τ}:=\{α\in (0;1): ||qα||\geq\fracγ{q^τ}\quad\forall q\in\Bbb{N}\},\qquad||x||:=\inf_{p\in\Bbb{Z}}|x-p|. $$ In this paper we study the topology of these sets and we show that, for large $τ$ and for almost all $γ>0$, $D_{γ,τ}$ is a Cantor set.

Intrinsic Diophantine Approximation on General Polynomial Surfaces

We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $ψ$-approximable points on a curve, surface, etc., given as a graph of integer valued polynomials. We obtain complete answers to these questions for algebraically "nice" manifolds. This generalizes earlier work done in the case of curves.

Diophantine equations in separated variables and lacunary polynomials

We study Diophantine equations of type $f(x)=g(y)$, where $f$ and $g$ are lacunary polynomials. According to a well known finiteness criterion, for a number field $K$ and nonconstant $f, g\in K[x]$, the equation $f(x)=g(y)$ has infinitely many solutions in $S$-integers $x, y$ only if $f$ and $g$ are representable as a functional composition of lower degree polynomials in a certain prescribed way. The behaviour of lacunary polynomials with respect to functional composition is a topic of independent interest, and has been studied by several authors. In this paper we utilize known results and dev…

Arithmetic properties of polynomial solutions of the Diophantine equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.

分离变量与多项式幂和中的丢番图方程

Diophantine equations in separated variables and polynomial power sums

本文研究形如 $f(x)=g(y)$ 且多项式为幂和的丢番图方程，利用 Bilu 和 Tichy 的有限性准则证明，在适当假设下具有有界分母的无穷多有理数解仅在平凡情形下成立。

Vertical shift and simultaneous Diophantine approximation on polynomial curves

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the d…

On Diophantine equations over $\mathbb Z[i]$ with $52$ unknowns

In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.

Isolated points of Diophantine sets

Let $γ\in(0;\frac{1}{2}),τ\geq 1$ and define the "$γ,τ$ Diophantine set" as: $$D_{γ, τ}:=\{α\in (0;1): ||qα||\geq\fracγ{q^τ}\quad\forall q\in\Bbb{N}\},\qquad ||x||:=\inf_{p\in\Bbb{Z}}|x-p|.$$ We analyze the topology of these sets and we show that generally they have isolated points.

On large $F$-Diophantine sets

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. In addition, we show…

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.

孤立丢番图数

Isolated Diophantine numbers

该研究探讨丢番图数的拓扑结构，构造出同丢番图常数下的孤立丢番图数显式例子，证明丢番图集未必是康托集，并简述了这类孤立数的一般性质。

Exceptional sets for Diophantine inequalities

We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c.

Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. …

Liminf Sets in Simultaneous Diophantine Approximation

Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied.

丢番图元组与移位幂的乘积集

Diophantine tuples and product sets in shifted powers

本文研究了具有D_k(n)性质的丢番图元组，改进了多个已有结果，并探讨了其在非平凡移位幂集中的应用，结合筛法、丢番图逼近和极值图论思想。

缓慢的多项式混合、动力学Borel-Cantelli引理与动力学丢番图集的Hausdorff维数

Slow polynomial mixing, dynamical Borel-Cantelli lemma and Hausdorff dimension of dynamical diophantine sets

本文针对缓慢的多项式混合系统，建立了动力学Borel-Cantelli引理与特定动力学丢番图集Hausdorff维数的最优性结果。

A Quantitative Result on Diophantine Approximation for Intersective Polynomials

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomials structures in sumsets and limitations of the method.

多项勒让德多项式在丢番图逼近中的应用

Multiple Legendre polynomials in Diophantine approximation

本文通过构造多重勒让德多项式，改进了有理数对数由代数数逼近的上界，特别提高了log 2的非二次性指数上界。

通过多元多项式生成的准随机超图研究有限域上的$f$-丢番图集

$f$-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials

本文通过多元多项式构建准随机超图，系统研究有限域上的$f$-丢番图集，提出统一框架，推导了$k$-丢番图$m$-元组的渐进行为，改进了Chung和Graham的经典估计。

关于涉及初等对称多项式的丢番图方程及单位分解

On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity

本文研究了初等对称多项式值相等的丢番图方程，证明了在特定条件下解的数量有限或趋于无穷。

Ramsey properties of nonlinear Diophantine equations

We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters betaN, grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers *N.

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.

A note on Diophantine systems involving three symmetric polynomials

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $σ_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} σ_{i}(\bar{X}_{2n})=a, \quad σ_{2n-i}(\bar{X}_{2n})=b, \quad σ_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. …

Polynomial parametrization of the solutions of Diophantine equations of genus 0

Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients.

计数有理数和丢番图逼近在缺失数字康托集中的应用

Counting rationals and diophantine approximation in missing-digit Cantor sets

本文通过傅里叶ℓ^1维度方法，研究了缺失数字康托集中的有理数计数和丢番图逼近问题，并对多个猜想做出了进展。

丢番图集

Diophantine sets

本文研究了丢番图集在交换环中的推广，并证明了关于有限呈现方案的覆盖性质。

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.

On polynomial-time solvable linear Diophantine problems

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, finds a nonnegative integer solution to the system Ax=b or determines that no such solution exists, provided that b is located sufficiently "deep" in the cone generated by the columns of B. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.

On the finiteness of solutions for polynomial-factorial Diophantine equations

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exists only finitely many $l$ such that $l!$ is represented {by} $N_A(x)$, where $N_A$ is a norm form constructed from the field norm of a field extension $K/\mathbf Q$. We also deal with the equation $N_A(x)=l!_S$, where $l!_S$ is the Bhargava factorial. In this paper, we also show that the Oesterlé-Masser conjecture implies that for any infinite subset $S$ of $\mathbf Z$ and for any polynomial …

Some Rational Diophantine Sextuples

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. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.

The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an app…

关于由LLL算法得到的联立丢番图逼近质量的注记

A note on the quality of simultaneous Diophantine approximations obtained by the LLL algorithm

本文针对LLL算法所得联立丢番图逼近的质量问题，对Bosma和Smeets的相关结果作出了显著改进。

An equivalence principle between polynomial and simultaneous Diophantine approximation

We show that Mahler's classification of real numbers $ζ$ with respect to the growth of the sequence $(w_{n}(ζ))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(λ_{n}(ζ))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximatio…

Triples which are $D(n)$-sets for several $n$'s

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set. $D(1)$-sets are known as simply Diophantine $m$-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine $m$-tuple ($D(1)$-set) which is also a $D(n)$-set for some $n\neq 1$. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophanti…

Diophantine Approximations on Definable Sets

Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structure which give a general framework to work with sets mentioned above. It complements the theorem of Pila-Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We focus our attention on polynomially bounded o-minimal structures, allow algebraic points of bounded …

从球体到开集的维度质量转移原理及其在动力学迪奥菲尼问题中的应用

A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

本文提出了一种从球体到开集的维度质量转移原理，用于解决动力学迪奥菲尼逼近中的缩小目标问题，并揭示了热力学正式主义与该领域的新联系。

A note on the Hausdorff dimension of some liminf sets appearing in simultaneous Diophantine approximation

Let Q be an infinite set of positive integers. Denote by W_{τ, n}(Q) (resp. W_{τ, n}) the set of points in dimension n simultaneously τ--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{τ, n}\W_{τ, n}(Q) is established when n>1 and τ>1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed.

Bohr sets and multiplicative diophantine approximation

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.

圆法的一种极简版本与薄集上的丢番图问题

A minimalist version of the circle method and Diophantine problems over thin sets

本文提出圆法的极简版本，证明均值估计、Weyl 型估计及容许剩余类上的分布性质即可建立华林问题等加性问题的渐近公式，并将其推广到椭圆薄集等薄集上。