Igor E. Shparlinski
We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine $m$-tuples in finite fields, which is due to A. Dujella and M.Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).
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A Generalization of Diophantine Tuples
本文通过Shparlinski方法研究有限域中Diophantine tuples的推广，并获得了关于此类元组数量的节省幂结果。
Bipartite Diophantine tuples and their applications
本文研究了广义狄利克雷元组的二元变种及其应用，推广了Bugeaud--Dujella关于特定二元狄利克雷元组家族的结果，并解决了第二作者提出的相关问题，同时建立了二元狄利克雷元组与几种已知狄利克雷元组变种之间的新联系。
In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for odd prime $p$ and guarantee the existence of a $k$-Diophantine m-tuple provided $p$ is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in $\mathbb{F}_p$ as well as an asymptotic formula for the number of $k$-Diophantine $k$-tuples.
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.
Let $K$ be an imaginary quadratic field and $ \mathcal{O}_K$ be its ring of integers. A set $\{a_1, a_2, \cdots,a_m\} \subset \mathcal{O}_K\setminus\{0\}$ is called a Diophantine $m$-tuple in $\mathcal{O}_K$ with $D(-1)$ if $a_ia_j -1 = x_{ij}^2$, where $x_{ij} \in \mathcal{O}_K$ for all $i,j$ such that $1 \leq i < j \leq m$. Here we prove the non-existence of Diophantine $m$-tuples in $\mathcal{O}_K$ with $D(-1)$ for $m > 36$.
For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring $\mathbb{Z}_p$ of $p$-adic integers, as well as its residue field $\mathbb{F}_p$.
For a prime p, a Diophantine m-tuple in $\mathbb{F}_p$ is a set of m nonzero elements of $\mathbb{F}_p$ with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number $N^{(m)}(p)$ of Diophantine m-tuples in $\mathbb{F}_p$ for m=2,3 and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically $N^{(m)}(p)=\frac{1}{2^{m \choose 2 }}\frac{p^m}{m!} + o(p^m)$, and also show that if $p>2^{2m-2}m^2$, then there is at least one Diophan…
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。
For non-zero integers $n$ and $k\geq2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $S = \{a_1,a_2,\ldots, a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j\leq m$. Define $M_k(n):= \sup\{|S| : S$ has property $D_k(n)\}$. In a recent work, the second author, S. Kim and M. R. Murty proved that $M_k(n)$ is $O(\log n)$, for a fixed $k$, as we vary $n$. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$, $M_k(n) \leq 3\,ϕ(k)\, \log n$, if $n$ is sufficiently larger than $k$.
Explicit constructions of Diophantine tuples over finite fields
本文研究了有限域上丢番图数组的最大规模 $M(q)$，通过显式构造方法改进了其下界，并以乘法因子提升了 Dujella 和 Kazalicki 的近期结果。
An absolute bound for generalized Diophantine tuples over polynomial rings
研究特征0代数闭域上多项式环中广义丢番图数组的大小，对于k≥18证明除必要例外族外|A|≤6，结合行列式准则、Mason-Stothers定理推广和组合零点定理。
A set of $m$ positive integers $\{a_1, a_2, \dots , a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i < j \le m$. In 2004 Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. In this paper we prove this conjecture.
Yet another $S$-unit variant of Diophantine tuples
证明满足任意两数之积为给定多项式在正整数$S$-单位处取值的整数三元组仅有有限个，基于Corvaja和Zannier的结果及Schmidt子空间定理。
Let n be a nonzero integer and assume that a set S of positive integers has the property that xy+n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| <= 400 then |S| <= 32, and if |n| > 400 then |S| < 267.81 log|n| (log log|n|)^2. The question whether there exists an absolute bound (independent on n) for |S| still remains open.
Given a finite set of primes $S$ and a $m$-tuple $(a_1,\dots,a_m)$ of positive, distinct integers we call the $m$-tuple $S$-Diophantine, if for each $1\leq i < j\leq m$ the quantity $a_ia_j+1$ has prime divisors coming only from the set $S$. For a given set $S$ we give a practical algorithm to find all $S$-Diophantine quadruples, provided that $|S|=3$.
Let $(a_1,\dots, a_m)$ be an $m$-tuple of positive, pairwise distinct, integers. If for all $1\leq i< j \leq m$ the prime divisors of $a_ia_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we estimate the number of $S$-Diophantine quadruples in terms of $|S|=r$.
Improved upper bounds on Diophantine tuples with the property $D(n)$
本文研究具有性质 $D(n)$ 的 Diophantine 元组最大规模 $M_n$，通过证明中间元素贡献为 $O(\log\log|n|)$，将上界改进为 $M_n\leq(2+o(1))\log|n|$。
Let $S$ denote a set of primes and let $a_1,\ldots,a_m$ be positive distinct integers. We call the $m$-tuple $(a_1,\ldots,a_m)$ an $S$-Diophantine tuple if $a_ia_j+1=s_{i,j}$ are $S$-integers for all $i\not=j$. In this paper, we show that no $S$-Diophantine quadruple (i.e~$m=4$) exists if $S=\{2,q\}$ with $q\equiv 3\; (\bmod\, 4)$ or $q<10^9$. For two arbitrary primes $p,q<10^5$ we gain the same result.
Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
研究移位乘法子群的乘法结构及其与加性组合学和丢番图方程理论的联系，提出新的结果，包括改进的上界和对Sárközy猜想的进展。
Prime-power Diophantine tuples
该研究将丢番图数组的间隙论证推广到素数幂情形，得到正约化 $D(\pm p^r)$ 数组的基数一致上界，明确了不同素数幂下的基数界限及最大基数的阶性质。
Multiplicatively reducible subsets of shifted perfect $k$-th powers and bipartite Diophantine tuples
本文通过建立与二部 Diophantine 元组的联系，将 Hajdu-Sárközy 关于平移 k 次幂集合乘法不可约性的结果有效化，证明 min{|A|,|B|} ≪ log|n| 并给出 |A||B| 的上界，显著改进了多项已有结果。
Rational Diophantine sextuples with strong pair
本文研究了有理丢番图多元组问题，通过引入强对条件，成功构造出包含强丢番图对的有理丢番图六元组的无限族。
For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously D(n_1)-quadruples and D(n_2)-quadruples with distinct non-zero squares n_1 and n_2.
In this paper, we give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine $m$-tuples with the property $D(4)$. We also confirm the conjecture of the uniqueness of such an extension in some special cases.
Diophantine m-tuples of Triangular Numbers
本文研究由三角数构成的D(n)丢番图元组，证明不存在D(1)三角四元组，提出生成D(1)三角三元组的算法，利用广义佩尔方程设计高效判定算法，还得出相关数论结论并推进了判据研究。
On Diophantine $m$-tuples related to primitive elements of finite fields
本文研究与有限域本原元相关的丢番图\(m\)元组，给出\(\mathbb{F}_q\)上\(\mathcal{P}\)-丢番图\(m\)元组数量\(N_m\)的渐近公式，并证明当\(q\)足够大时存在这样的\(m\)元组。
On Diophantine graphs
本文定义并研究以正整数为顶点、乘积加一为平方数时连边的 Diophantine 图，给出其可扩张性、最大边数及色数界的结果。
Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$
本文证明每个 Diophantine 数组可表示为二次域整理想的范数乘以常数，并在互素情形下给出显式构造。
Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.
This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(α)$, where $S(α)$ is the set of all integers of the form $\lfloor αn^2 \rfloor$ for $n\geq α^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(α)$, the set of all such tuples, is infinite for all $α\in (0,1)\cap \mathbb{Q}$ and for almost all $α\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $α>0$ such that $T(α)$ is finite, then there is no perfect Euler brick. W…
The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $<x_1,...x_k>$ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometri…
$f$-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。
A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…
On Regular Higher Power Rational Diophantine Triples
本文研究了正则高次幂有理数迪菲安特三元组，构造了非平凡的无限族，并证明了当k=4时存在无限多个正元素三元组，还简要讨论了k=6和k=8的情况。
Diophantine stability for curves over finite fields
本文综述有限域上的丢番图稳定曲线，推导其一般性质并分析多个稳定曲线族。
We study some diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.
For a tuple $(θ_1,..,θ_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by $θ_1, ..., θ_M$ over the field of rational numbers from the approximability of the point $θ=(1,θ_1,...,θ_M)$ in projective space by hypersurfaces. The first given result is an new proof of an algebraic independence criterion, that was formerly proved by Laurent and Roy, and generalizes the Philippon criterion by introducing also evaluations of derivatives of global sections. …
Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
A Diophantine $m$-tuple with elements in the field $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$ be a threefold. Its $K$-rational points parametrize Diophantine triples over $K$ such that the product of the elements of the triple that corresponds to the point $(x,y,z,k)\in X(K)$ is equal to $k$. We denote by $\overline{X}$ the projective closure of $X$ and for a fixed $k$ by $X_k$ a variety defined by the same equation as $X$. We prove that the variety…
Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution …
$F$-Diophantine sets over finite fields
本文研究有限域上的 $F$-丢番图集，提出基于单项式展开性质的构造策略，并在 $F=x_1x_2\ldots x_k+1$ 情形下得到大小 $\gg_k \log q$ 的 $k$-丢番图元组，显著改进已有下界。
Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v$ that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with finite norms.
On certain $D(9)$ and $D(64)$ Diophantine triples
研究关于$D(9)$和$D(64)$丢番图三元组，推广刻画特定$D(l^2)$-丢番图三元组对应佩尔方程解的结果，利用该结果及代数数对数线性形式的界，对特定形式的$D(9)$和$D(64)$-丢番图三元组进行分类。
As it follows from the theory of almost periodic functions the set of integer solutions $q$ to the Kronecker system $|ω_{j} q - θ_{j}| < \varepsilon \pmod 1$, $j=1,\ldots,m$, where $1,ω_{1},\ldots,ω_{m}$ are linearly independent over $\mathbb{Q}$, is relatively dense in $\mathbb{R}$. The latter means that there is $L(\varepsilon)>0$ such that any segment of length $L(\varepsilon)$ contains at least one integer solution to the Kronecker system. We give some lower and upper non-effective (asymptotic) estimates for $L(\varepsilon)$ and, in particular, show that $L(\varepsilon) = \left(\frac{1}{\v…
The height of a rational number $p/q$ is denoted by $h(p/q)$ and equals $\text{max}(|p|,|q|)$ provided p/q is written in lowest terms. The height of a rational tuple $(x_1,...,x_n)$ is denoted by $h(x_1,...,x_n)$ and equals $\text{max}(h(x_1),...,h(x_n))$. Let $G_n={x_i+1=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}$. Let $f(1)=1$, and let $f(n+1)=2^{(2^{(f(n))})}$ for every positive integer n. We conjecture: (1) if a system $S \subseteq G_n$ has only finitely many solutions in rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \…
We show that the equation in the title (with $Φ_m$ the $m$th cyclotomic polynomial) has no integer solution with $n\ge 1$ in the cases $(m,p)=(15,41), (15,5581),(10,271)$. These equations arise in a recent group theoretical investigation by Z. Akhlaghi, M. Khatami and B. Khosravi.
Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is strictly increasing, (2) if a non-decreasing function g from positive integers to positive integers satisfies f(n) \geq g(n) for any n, then a finite-fold Diophantine representation of g does not exist, (3) if the question of the title has a positive answer, then there is a computable strictly increasing function g from positive integers to positive integers su…
Diophantine $D(n)$-quadruples in $\mathbb{Z}[\sqrt{4k + 2}]$
本文研究了二次整数环 $\mathbb{Z}[\sqrt{d}]$ 中的 Diophantine $D(n)$-四元组，在特定条件下证明了对于满足特定同余关系的 $n$，存在无穷多个 $D(n)$-四元组，并给出了 $d=10$ 时的实例。
In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{*}$ for Laurent series over a finite field. We prove that for an integer $n\geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j\geq 1}$ and a sequence of algebraic Laurent series $(ξ_j)_{j\geq 1}$ such that deg $ξ_j=p^{k_j}+1$ and \begin{equation} w_1(ξ_j)=w_1 ^{*}(ξ_j)=\ldots =w_n(ξ_j)=w_n ^{*}(ξ_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $ξ$ for which $w_n(ξ)$ and $w_n^{*}(ξ)$ are different.
It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of R^n that is invariant under rational translations and which does not have full Lebesgue measure with an the closure of an open set of positive measure cannot be positive and finite. Analogues for $p$-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation.
We prove the following function field analog of the Hardy-Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n,r be positive integers and q an odd prime power. For distinct polynomials a_1, ..., a_r over F_q of degree <n let π(q,n;a) be the number of monic polynomials f over F_q of degree n such that f+a_1, ..., f+a_r are simultaneously irreducible. We prove that π(q,n;a) asymptotically equals q^n/n^r as q tends to infinity on odd prime powers and n,r are fixed (the tuple a1,...,a_r need not be fixed).
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite sets.
Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rat…
For a positive integer n, let θ(n) denote the smallest positive integer b such that for each system S \subseteq {x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}} which has a solution in positive integers x_1,...,x_n and which has only finitely many solutions in positive integers x_1,...,x_n, there exists a solution of S in ([1,b] \cap N)^n. We conjecture that there exists an integer δ \geq 9 such that the inequality θ(n) \leq (2^{2^{n-5}}-1)^{2^{n-5}}+1 holds for every integer n \geq δ. We prove: (1) for every integer n>9, the inequality θ(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-…
Diophantine stability for elliptic curves on average
本文证明正密度的秩1椭圆曲线在素数处具有丢番图稳定性，并探讨其对希尔伯特第十问题的影响。
S. Lang conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Noguchi. We also deal with the finiteness property of $S$-units points of those Diophantine equations over number fields.
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.
We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in fewer than exp((4+o(1))s^{1/2}(log s)^{-1/2}) proper linear subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for m=2 [Compositio 36 (1988), 37-56]. Furthermore we prove that for any algebraic number field K of degree n, any integer m with 1<=m<n, and any sufficiently large s there are integers b_0,...,b_m in a number field which ar…
Let $\mathbb F_q$ be a finite field, $F/\mathbb F_q$ be a function field of genus $g$ having full constant field $\mathbb F_q$, $\mathcal S$ a set of places of $F$ and $H$ the holomorphy ring of $\mathcal S$. In this paper we compute the density of coprime $m$-tuples of elements of $H$. As a side result, we obtain that whenever the complement of $\mathcal S$ is finite, the computation of the density can be reduced to the computation of the $L$-polynomial of the function field. In the rational function field case, classical results for the density of coprime $m$-tuples of polynomials are obtain…
Recently, W. M. Schmidt and L. Summerer introduced a new theory which allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and to discover new ones. They first note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point. Then they prove that the $n$-tuple of these successive minima can in turn be approximated up to bounded difference by a function from a certain class. In this paper, we show that the same is true within a sma…
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