Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $ψ$ let $\cal C(ψ)$ denote the set of simultaneously $ψ$--approximable points lying on $\cal C$. We show that $\cal C$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\cal C$ of $\cal C(ψ)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\cal C$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $ψ$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\cal C(ψ)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples.
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in …
In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich-Zorin in the divergence case.
We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.
A set of $m$ distinct nonzero rationals $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+1$ is a perfect square for all $1\leq i<j\leq m$, is called a rational Diophantine $m$-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.
This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdor? dimension results for the sets of simultaneously well-approximable points on planar curves, established in [2, 6, 10, 20].
We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.
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…
The convergence theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diophantine approximation on planar curves and the distribution of rational points' (by V. Beresnevich, H. Dickinson, and S. Velani, with an Appendix: `Sums of two squares near perfect squares' by R.C. Vaughan, To appear: Annals of Math., Pre-print: arkiv:math.NT/0401148, (2004), 1-52.) and thereby completes the general metric theory for planar curves.
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and Dujella, Kazalicki, Mikic and Szikszai recently proved that there exist infinitely many rational Diophantine sextuples. In this paper, generalizing the work of Piezas, we describe a method for generating new parametric formulas for …
The inhomogeneous metric theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann. of Math. (2), 166 (2007), pp. 367-426] and [Invent. Math., 166 (2006), pp. 103-124]. The key lies in obtaining essentially the best possible results regarding the distribution of `shifted' rational points near planar curves.
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that k…
We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.
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.
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.
In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.
We study the ``imaginary" binary quadratic form equations ax^2+bxy+cy^2+g=0 over k[t] in rational function fields, showing that a condition with respect to the Artin reciprocity map, is the only obstruction to the local-global principle for integral solutions of the equation.
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.
Quartic Rational Diophantine Quadruples and the Euler Surface
University of Zagreb, Faculty of Civil Engineering（萨格勒布大学土木工程学院）；University of Zagreb, Faculty of Science（萨格勒布大学理学院）；University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）
In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operati…
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…
In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are also proved in the inhomogeneous setting. Applications of the main theorem include the Khintchine-Jarník type theorem for divergence for arbitrary non-degenerate curve in $\mathbb{R}^n$.
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.
Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points $(x,y)$ on the curve which satisfy simultaneously the inequalities $\| q x \| < ψ_1(q)$ and $\| qy \| < ψ_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < ψ(q)$, we establish a Hausdorff measure convergence re…
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…
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.
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.
Duffin and Schaeffer have generalized the classical theorem of Khintchine in metric Diophantine approximation in the case of any error function under the assumption that all the rational approximants are irreducible. This result is extended to the case where the numerators and the denominators of the rational approximants are related by a congruential constraint stronger than coprimality.
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study o…
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 $g$.
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 $f$.
A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of $ψ$-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obt…
We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (…
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of $m\times n$ matrices over $\Bbb R$ is introduced and studied. This notion generalises the one of non-planarity in $\Bbb R^n$ and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weig…
We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler's generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue--Mahler for degree $3$ implies the theorem of Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line $¶^1…
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) \…
In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using another height function of a rational number in $\mathcal{K}$, i.e. the denominator obtained from its periodic 3-adic expansion, a complete metric theory for this variant intrinsic Diophantine approximation is presented which yields the divergence theory of Mahler's original question.
One of Legendre's theorems on the Diophantine equation $ax^2+by^2+cz^2=0$ provides necessary and sufficient conditions on the existence of nonzero rational solutions of this equation, which helps determine the existence of rational points on a conic. In this paper, we provide a new proof of this famous theorem using Hasse invariant and Jacobi symbol from the theory of quadratic forms.
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that a…
We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.
We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.
We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over that finite field. Both proofs depend upon Matijasevich's theorem.
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.
In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known when $k=2$ or $3$, till now only one numerical solution of the equation in positive rational numbers has been published when $k=4$, and no nontrivial solution is known when $k \geq 5$. We describe a method of generating infinitely many positive rational solutions of the equation when $k=4$. We use the positive rational solutions of the equation with $k=2,\,…
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.
The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to $C^1$ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar cu…
We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3. These results rely on exhibiting families of positive-rank elliptic curves.
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…
