Manfred K"\uhleitner, Werner Georg Nowak
This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to write m as a sum of two squares of integers.
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We consider some asymptotic analysis for series related to the work of Hardy and Littlewood on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH.
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
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…
A paradigm for a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. This is achieved through the study of subgroups of nonstandard models of Dedekind domains called diophantine approximation groups. The arithmetic of diophantine approximation groups is defined in a way which extends the ideal-theoretic arithmetic of algebraic number theory, using the structure of an approximate ideal: a bi-filtration by subgroups along which partial products may be performed.
In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see t…
We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using ubiquitous systems. For $S$ consisting of more than one prime, finite or infinite, the divergence results are new even in the homogeneous setting. We also prove the convergence case of the theorem, including in particular, the $S$-arithmetic inhomogeneous counterpart of the Baker-Sprindžuk conjectures using nondivergence estimates for flows on homogeneous spaces i…
Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of the situation with integer numbers is incorrect. In many situations, we have to utilize different rules of counting and operating. This is a consequence of the…
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.
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.
Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space.
Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $Γ_{p,a}\subset SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an estimate for the traces of a set of generators for $Γ_{p,a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a g…
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。
Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry
本文综述拟射影簇上 adelic 线丛的基础、算术正性及其在均匀分布定理和一致 Bogomolov 猜想中的应用。
This paper has been withdrawn by the author because Conjecture 1 is false. Please see arXiv:0901.2093 for a justification that Conjecture 1 is false. The other main results are also available from the above URL.
Some Diophantine equations involving arithmetic functions and Bhargava factorials
本文将 Luca 关于阶乘与 Euler 函数、除数和及除数计数函数的丢番图方程有限解结论，推广到多阶乘乘积和除数 k 次幂和函数，并扩展至 Bhargava 阶乘。
Arithmetic statistics and diophantine stability for elliptic curves
本文研究有理数域椭圆曲线在素数幂循环扩张中的Mordell-Weil群与Tate-Shafarevich群稳定性，提出“Sha”-稳定性概念，并利用算术统计与岩泽理论证明非CM椭圆曲线在几乎所有素数处稳定。
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of M$\operatorname{\ddot{o}}$bius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fractions expansions.
This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…
Arithmetic holonomy bounds and effective Diophantine approximation
本研究通过量化算术和乐界提出射影直线与乘法群上有效丢番图逼近新方法，推导出代数数高阶根的有效无理性度量，可导出经典有效丢番图定理并给出多个数论应用。
In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham.
On Transformation properties of hypergeometric motives and Diophantine equations
本文研究高斯超几何函数变换公式在超几何动机上的算术类比，并将其应用于某些丢番图方程的研究。
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.
In this paper we study elliptic curves which have a number of points whose coordinates are in arithmetic progression. We first motivate this diophantine problem, prove some results, provide a number of interesting examples and, finally point out open questions which focus on the most interesting aspects of the problem for us.
Analytic expressions pertaining to certain arithmetical functions
本文提出一种方法，为两类丢番图方程正整数解上的算术和推导解析表达式。
S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.
In this paper, first, we prove that the Diophantine system \[f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)\] has infinitely many integer solutions for $f(X)=X(X+a)$ with nonzero integers $a\equiv 0,1,4\pmod{5}$. Second, we show that the above Diophantine system has an integer parametric solution for $f(X)=X(X+a)$ with nonzero integers $a$, if there are integers $m,n,k$ such that \[\begin{cases} \begin{split} (n^2-m^2) (4mnk(k+a+1) + a(m^2+2mn-n^2)) &\equiv0\pmod{(m^2+n^2)^2},\\ (m^2+2mn-n^2) ((m^2-2mn-n^2)k(k+a+1) - 2amn) &\equiv0 \pmod{(m^2+n^2)^2}, \end{split} \end{cases}\] where $k\equiv0\pmod{4}$ when…
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effect…
Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_σ: y^2 = x (x+σp)(x+σq) . (σ=\pm 1). E=E_σis also denoted as E_+ or E_- when σ= +1or $-1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(\sqrt -1) (and over the rational field Q is determined in several cases; and results on solutions of related Diophantine equations and simultaneous Pellian equations will be given. The arithmetic constructs over Q of the elliptic curve E have been studied in the last paper1, the Selmer groups are determined, results on Mordell-Weil group,…
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。
Arithmetic in the Boij Söderberg Cone
本文通过将猜想转化为Boij-Söderberg锥体中的算术问题，研究了格雷码模的Betti数下界，并在五重和六重情况下证明了新的结果，揭示了Betti表、交换代数和丢番图方程之间的联系。
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
This paper deals with three consequences of the existence of Arithmetic Teichmuller spaces of arXiv:2106.11452. Let $\mathscr{X}_{F,\mathbb{Q}_p}$ (resp. $B=B_{\mathbb{Q}_p}$) be the complete Fargues-Fontaine curve (resp. the ring) constructed by Fargues-Fontaine with the datum $F={\mathbb{C}_p^\flat}$ (the tilt of $\mathbb{C}_p$), $E=\mathbb{Q}_p$. Fix an odd prime $\ell$, let $\ell^*=\frac{\ell-1}{2}$. The construction (§7) of an uncountable subset $Σ_{F}\subset \mathscr{X}_{F,\mathbb{Q}_p}^{\ell^*}$ with a simultaneous valuation scaling property (Theorem 7.8.1), Galois action and other symm…
In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the Diophantine classes and numbers admitting approximations by rational numbers at a given 'speed'. We also show that irrational numbers with finite upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.
In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.
Diophantine Approximation in local function fields via Bruhat-Tit trees
基于 Bruhat-Tits 树的算术商理论，获得了全局函数域上丢番图逼近的 Dirichlet 风格定理，并计算了相关常数与逼近集合测度。
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.
In this paper, we prove an asymptotic formula for the average number of solutions to the Diophantine equation $axy-x-y=n$ in which $a$ is fixed and and $n$ varies.
In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theore…
We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Furthermore, we show that the Lang-Waldschmidt conjecture implies a special case of Vojta's conjecture with truncation in arbitrary dimension. O…
This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.
From Hilbert's Tenth Problem to Quantum Speedup: Explicit Oracles for Bounded Diophantine Systems
本文提出了一种完全可逆的量子算法框架，用于解决有界整数域上的任意多项式丢番图方程，通过显式门级合成评估 oracle 实现幅度放大，从而在空间复杂度和非克莱因门深度上实现了高效的量子算术处理，保证了与经典穷举搜索相比的二次加速。
One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.
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.
We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.
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。
A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.
Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$ form a cocompact discrete subgroup $Γ_{p,a}\subset SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an estimate for the traces of a set of generators for $Γ_{p,a}$. Empirical results summarized in several tables show that the trace has significant and irregular fluctuations which is reminiscent of the behavior of the size of a g…
A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester's sequence, Giuga numbers, Znám's problem, the inheritance problem, and Curtiss's bound on solutions of a unit fraction equation. Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable $7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a cond…
Diophantine approximation with integers having no large prime factors
研究无大素因子整数n与无理数α的逼近问题，证明对于任意θ<6/17，存在无限多个y-光滑数n满足‖nα‖ <n^{-θ}，改进了Baker和Yau的先前结果。
On the exponential Diophantine equation ${\displaystyle p\cdot 3^{x}+p^{y}=z^2}$ with $p$ a prime number
本文利用模算术性质研究素数 $p$ 下的指数丢番图方程 $p\cdot 3^x+p^y=z^2$，分类给出不同同余情形的唯一解或无穷解集，并讨论无解情形与开放问题。
Fermat descent
本文通过商叠层视角重新阐释下降理论，聚焦于广义费马方程本原整数解相关叠层的算术性质研究，而非求解该方程的具体实例。
Average orders of Goldbach Estimates in Arithmetic Progressions
本文研究了算术级数中Goldbach表示的平均数问题，获得了渐近结果，并证明了该结果本质上是最佳的。
We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.
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}}. If Matiyasevich's conjecture on finite-fold Diophantine representations is true, then for every computable function f:N->N there is a positive integer m(f) such that for each integer n>=m(f) there exists a system S \subseteq E_n which has at least f(n) and at most finitely many solutions in integers x_1,...,x_n. This conclusion contradicts to the author's conjecture on integer arithmetic, which implies that the heights of integer solutions to a Diophantine equation are computably bounded, if these solutions form a finite set…
We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable range for $q$.
On arithmetic terms expressing the prime-counting function and the n-th prime
本研究提出首个固定长度的初等闭式算术项，可分别表示素数计数函数$\pi(n)$与第$n$个素数$p(n)$，仅用有限固定次数的初等算术运算，为素数是否存在次序的基础问题提供构造性解答。
Diophantine conditions in the law of the iterated logarithm for lacunary systems
本文研究缺项和 $\sum f(n_k x)$ 满足重对数律所需的丢番图条件，证明节省对数阶因子是最优条件，并揭示重对数律比中心极限定理要求更强的算术条件。
Weighted averages of arithmetic functions over regular integers modulo $n$
该研究推广了Kiuchi与Matsuoka的恒等式，建立了算术函数部分和的一般变换公式，将其应用于经典乘性函数的加权平均，得到了显式渐近公式，为推导渐近结果提供了统一框架。
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