Yuri Matiyasevich, Zhi-Wei Sun
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.
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In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.
It is an old problem in the area of Diophantine definability to determine whether $\mathbb{Q}$ is Diophantine in $\mathbb{Q}(z)$. We provide a positive answer conditional on two standard conjectures on elliptic surfaces.
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.
Undecidability on Diophantine equations over $\mathbb Z[i]$ with $20$ unknowns
本文证明含20个未知数的整系数多项式方程在Gauss整数环上的可解性不存在判定算法，将此前52个变量的不可判定性纪录改进至20个。
We show that the Diophantine pair $\{1, 3\}$ can not be extended to a Diophantine quintuple in $\mathbb{Z}\left[\sqrt{-2}\right]$. This result completes the work of the first author and establishes non-extensibility of the Diophantine pair $\{1, 3\}$ to a Diophantine quintuple in $\mathbb{Z}\left[\sqrt{-d}\right]$ for all $d\in \mathbb{N}$.
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.
Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine when $K$ is a number field. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.
On Iwasawa theory of abelian varieties over $\mathbb{Z}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions
本文研究Abel曲线上Iwasawa理论，探讨丢番图稳定性及整数丢番图扩张，并改进Mazur猜想相关结果，扩展至具有良好超奇异减的椭圆曲线。
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 $\mathbb{Z}_p$.
On Some Diophantine Parameters of the Cyclic Torsion Subgroups of Odd Order of Elliptic Curves over $\mathbb{Q}$
本文研究有理数域上椭圆曲线奇阶循环挠子群，并给出了其若干显式丢番图参数。
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.
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 $\mathbb{F}_p$.
On Diophantine equations over the integer rings of quadratic fields
本文针对二次域整数环$O_K$，证明其16未知量多项式方程求解问题不可判定，实二次域时仅需15个未知量，完善了希尔伯特第十问题相关不可判定性结论。
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $pγ$, $p$ running over the primes and $γ$ being a fixed irrational, for Gaussian primes.
$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with $7$ unknowns
该研究改进了有理数集减去整数集在有理数域上为丢番图集的未知量个数，从10个降至7个并推广到任意整体域，结合前期结果证明了相关判定问题不存在算法。
We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory st…
The Diophantine problem for addition and divisibility for rings of $S$-integers of quadratic imaginary extensions of $\mathbb{Q}$
本文证明对于 $\mathbb{Q}$ 的二次虚扩张的 $S$-整数环（$S$ 非空），加法与整除性的丢番图问题不可判定，与整数环情形（可判定）形成对比。
Exact Diophantine approximation$\colon$ the simultaneous case in $\mathbb{R}^{2}$
该研究填补了Fregoli关于$\mathbb{R}^{2}$中精确逼近阶集合豪斯多夫维数研究的空白，完善了联立情形下精确丢番图逼近的相关理论。
Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ are prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $…
We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of $\mathrm{SL}_2 ( \mathcal{O}_K )$ on the punctured complex plane $\mathbb{C}^2 \setminus \{ \mathbf{0} \}$, where $K$ is a number field whose ring of integers $\mathcal{O}_K$ is discrete and within a unit distance of any complex number. The results are similar to those of Laurent and Nogueira for $\mathrm{SL}_2 ( \mathbb{Z} )$ action on $\mathbb{R}^2 \setminus \{ \mathbf{0} \}$ albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.
In recent years, the ergodic theory of group actions on homogeneous spaces has played a significant role in the metric theory of Diophantine approximation. We survey some recent developments with special emphasis on Diophantine properties of affine subspaces and their submanifolds.
In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.
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…
Mixed quantifier prefixes over Diophantine equations with integer variables
该研究回顾希尔伯特第十问题，并分析整数变量丢番图方程的混合量词前缀，证明 ∀²∃⁴ 及若干受界全称量词情形不可判定，且提出 ∀²∃² 不可判定猜想。
We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline{y}$ in $\mathbb{F}_q((T^{-1}))^2$ by the $SL_2(\mathbb{F}_q[T])$-orbit of a given point $\underline{x}$ in $\mathbb{F}_q((T^{-1}))^2$.
Regularity of Diophantine quadruples over $\mathbb{Q}(i)[X]$
该研究证明了$\boldsymbol{\text{Q}}(i)[X]$上至少含一个非常数多项式的丢番图四元组均为正则的，其正则性条件为$(a+b-c-d)^2=4(ab+1)(cd+1)$，该结果与部分域上的对应结论一致，与另一域上的情形相反。
$\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$ with 32 unknowns
本文证明 $\mathbb Q\setminus\mathbb Z$ 在 $\mathbb Q$ 上是丢番图的，仅需 32 个未知数，并结合 Sun 的结果得出相关判定问题不可判定。
On exotic Diophantine triples in $\mathbb{R}[X]$
研究实数域上多项式环\(\mathbb{R}[X]\)中奇异丢番图三元组问题，通过证明得出若\(a,b,c\)都不等于\(1\)且至少有一个非常数元素时不存在该三元组，即含非常数元素的此类三元组必含\(1\)。
Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic $0$ and unramified mixed characteristic. It also encompasses the conditional existential decidability results known for $\mathbb{F}_p(\!(t)\!)$ and its finite extensions, due to Denef-Schoutens. On the other hand, it also applies to the setting of infinite ramification, providing us with an abundance of infinitely ramified extensions of $\mathbb{Q}_p$ and $\mathbb{F}_…
We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group $G$. If $G$ has Hausdorff dimension $Q$, we show that its Diophantine exponent is equal to $(Q+1)/Q$, generalizing the case $G=\mathbb R^n$. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group $\mathbb H^n$, distinguishing between two notions of Diophantine approximation by rational points in $\mathbb H^n$: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantin…
In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrtε)/\mathbb{Q}(\sqrt{d})$ (where $ε$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some "large image" results due to Ellenberg regarding images of Galois representations coming from $\mathbb…
We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for ${\mathbb Z}$ in ${\mathbb Q}$, provided one assumes a strong variant of the Bombieri-Lang …
We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.
Arithmetic of elliptic curves induced by regular Diophantine triples
本文研究正则丢番图三元组诱导的椭圆曲线的算术，明确其挠子群结构，给出二次域上获3阶点的判定准则并验证特定族无此情况，还分析另一类D(-k²)三元组诱导曲线的挠与秩。
The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namel…
Let $r\ge 1$ be an integer and ${\bf U}:=\{U_n\}_{n\ge 0}$ be the Lucas sequence given by $U_0=0,~U_1=1$, and $U_{n+2}=rU_{n+1}+U_n$ for $n\ge 0$. In this paper, we explain how to find all the solutions of the Diophantine equation, $AU_{n}+BU_{m}=CU_{n_1}+DU_{m_1}$, in integers $r\ge 1$, $0\le m<n,~0\le m_1<n_1$, $AU_n\ne CU_{n_1}$, where $A,B,C,D$ are given integers with $A\ne 0,~B\ne 0$, $m,n,m_1,n_1$ are nonnegative integer unknowns and $r$ is also unknown.
In this paper we provide criteria for the insolvability of the Diophantine equation $x^2+D=y^n$. This result is then used to determine the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$. We also determine some criteria for the divisibility of the class number of the quadratic field $\mathbb{Q}(\sqrt{-D})$ and this result is then used to discuss the solvability of the Diophantine equation $x^2+D=y^n$.
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.
We prove that for every integer $n$, there exist infinitely many $D(n)$-triples which are also $D(t)$-triples for $t\in\mathbb{Z}$ with $n\ne t$. We also prove that there are infinitely many triples with the property $D(-1)$ in $\mathbb{Z}[i]$ which are also $D(n)$-triple in $\mathbb{Z}[i]$ for two distinct $n$'s other than $n = -1$ and these triples are not equivalent to any triple with the property $D(1)$.
Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators
本文研究了平面直线上具有限制分母的乘法丢番图逼近问题，推导了收敛条件并给出了Hausdorff维度的上界。
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。
Reverse Engineered Diophantine Equations over $\mathbb{Q}$
本研究将整数完全幂有限子集的对应定理推广至有理完全幂，证明对任意有理完全幂的有限子集$S$，存在整系数多项式使其有理点集与有理完全幂的交集恰为$S$，方法依托广义费马方程求解结果与二元递推序列完全幂有限性结论。
Intrinsic Diophantine approximation on circles and spheres
本文提出统一框架，将六个圆和球面上的内蕴丢番图逼近的拉格朗日谱与实数域和复数域的谱关联，从而完整确定各谱的最小聚点及初始离散部分。
We study systems of polynomial equations in infinite finitely generated commutative associative rings with an identity element. For each such ring $R$ we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is reducible to the same problem in $R$. If, in particular, $R$ has positive characteristic or, more generally, if $R$ has infinite rank, then we further obtain a…
We study the problem of best approximations of a vector $α\in{\mathbb R}^n$ by rational vectors of a lattice $Λ\subset {\mathbb R}^n$ whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.
In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for which $y$ is not a sum of two consecutive squares. We also study the above equation with fixed $y$ and with fixed $q.$
Let $Θ= (θ_1,θ_2,θ_3)\in \mathbb{R}^3$. Suppose that $1,θ_1,θ_2,θ_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ α(Θ) = \sup \{γ>0:\,\,\, \limsup_{t\to +\infty} t^γψ_Θ(t) <+\infty \} ,$$ $$β(Θ) = \sup \{γ>0:\,\,\, \liminf_{t\to +\infty} t^γψ_Θ(t) <+\infty\} $$ we prove $$ β(Θ) \ge {1/2} ({α(Θ)}/{1-α(Θ)} +\sqrt{{α(Θ)}/{1-α(Θ)})^2 +{4α(Θ)}/{1-α(Θ)}}) α(Θ) $$
Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb N$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples $(a,b,c)\in \mathbb N^3$ with $c<b<a$ such that $ab+1,ac+1$ and $ab+1$ lie in the set $\mathcal R_g$. In this paper, we prove effective finitness results for the set $\mathcal D_g$.
In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $χ$ satisfying that the Frey elliptic curve representation twisted by $χ$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (toge…
Let $r, v, n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations $$ n=1^{r}\cdot |k_{1}|^{v}+2^{r}\cdot |k_{2}|^{v}+3^{r}\cdot |k_{3}|^{v}+\ldots, $$ for ${\bf k}=(k_1,k_2,k_3,\dots)\in\mathbb{Z}^{\infty}$. For each $(r,v)\in\mathbb{N}\times\{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.
We show the insolvability of the Diophantine equation $ax^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1 \pmod {12}$ and $b=2^da-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we investigate the more general family with $b=2^da-3^r$, where $r$ is a positive odd integer. As a consequence, we found an infinite family of hyperelliptic curves with trivial torsion over $\mathbb{Q}$. We conclude by providing some numerical evidence corroborating the main results.
A positive density of elliptic curves are diophantine stable in certain Galois extensions
本文证明，对于循环p-扩张L/Q（p=3或5），存在正密度的椭圆曲线E（按高度排序）在L中具有丢番图稳定性。
Matrix Solutions to the Diophantine equation $X^4+ Y^4=2Z^4$
本研究证明丢番图方程 $X^4+Y^4=2Z^4$ 在多种矩阵环及特定域扩张中存在无穷多个整数和有理数矩阵解，扩展了方程解的矩阵范围。
We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) = K_{n+1}(x_0,\ldots,x_n)K_{n+1}(x_1,\ldots, x_{n+1})$. We show that with certain restrictions on $P(x)$ the set of its solutions has a rich structure. In particular, we provide several ways of generating new solutions from the existing ones. In the last section we discuss the relation between the solutions of the above Diophantine equation for arbitrary values of $n$ an…
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…
This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $θ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $γ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(θ,γ)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle Nθ- γ\rangle|$. We show that for every $θ$ there exists $γ$ such that $c(θ,γ)\geq q^{-2}$, and find a sufficient condition on $θ$ which forces $c(θ,γ) \leq q^{-2}$ for every $γ$. Given $θ$, we prove that the set $BA_θ=\left\{γ\in\m…
Motivated by the theory of Diophantine $m$-tuples, we study rational points on quadratic twists $H^d:d y^2=(x^2+6x-18)(-x^2+2x+2)$, where $|d|$ is a prime. If we denote by $S(X)=\{ d \in \mathbb{Z}: H^d(\mathbb{Q})\ne \emptyset, |d| \textrm{ is a prime}\textrm{ and } |d| < X\},$ then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as $X \rightarrow \infty$ $$\frac{43}{256}+o(1)\le \frac{\#S(X)}{2π(X)}\le \frac{46}{256}+o(1).$$
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。
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。
We consider the question of how well points in a quadric hypersurface $M\subset\mathbb R^d$ can be approximated by rational points of $\mathbb Q^d\cap M$. This contrasts with the more common setup of approximating points in a manifold by all rational points in $\mathbb Q^d$. We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence be…
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