L. A. Gutnik
I present here the proofs of results, which are obtained in my papers "On the linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 1617-B96, pp. 1 - 23 (in Russian), and "On the systems of linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 2663-B96, pp. 1 - 18 (in Russian).
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The determination of Jacobi sums, their congruences and cyclotomic numbers have been the object of attention for many years and there are large number of interesting results related to these in the literature. This survey aims at reviewing results concerning the diophantine systems for finding the cyclotomic numbers and coefficients of Jacobi sums and to indicate the current status of the problem.
We investigate the solvability of the Diophantine equation $x^2-my^2=\pm p$ in integers for certain integer $m$ and prime $p$. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly bigger than $1$.
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.
The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss. In this paper we obtain an explicit expression for cyclotomic numbers of order 2l^2 in terms of the coefficients of the Jacobi sums of lower orders. At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.
By viewing the regular $N$-gon as the set of $N$th roots of unity in the complex plane we transform several questions regarding polygon diagonals into when a polynomial vanishes when evaluated at roots of unity. To study these solutions we implement algorithms in Sage as well as examine a trigonometric diophantine equation. In doing so we classify when a metallic ratio can be realized as a ratio of polygon diagonals, answering a question raised in a PBS Infinite Series broadcast. We then generalize this idea by examining the degree of the number field generated by a given ratio of polygon diag…
Moments of Higher Logarithmic Derivatives and Exceptional Zeros in Cyclotomic Fields
研究分圆域中与狄利克雷\(L\)函数相关的高阶对数导数\(\mathcal{L}^{(r)}(1,\chi)\)，通过证明算术公式和建立矩渐近性，得出足够大导体的素分圆域的第二李系数为正，排除斯塔克型无零区域中可能的例外零点。
We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI.
Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be …
This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.
We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of $K$-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Pethö. This paper extends the previous work of \cite{Patel}.
\noindent In this article, we determine all the integers $c$ having at least two representations as difference between two linear recurrent sequences. This is a variant of the Pillai's equation. This equation is an exponential Diophantine equation. The proof of our main theorem uses lower bounds for linear forms of logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.
This article deals with a conjecture generalizing the second case of Fermat's Last Theorem, called $SFLT2$ conjecture: {\it Let $p>3$ be a prime, $K:=\Q(ζ)$ the $p$th cyclotomic field and $\Z_K$ its ring of integers. The diophantine equation $(u+vζ)\Z_K=\mk w_1^p$, with $u,v\in\Z\backslash\{0\}$ coprime, $uv\equiv 0 \bmod p$ and $\mk w_1$ ideal of $\Z_K$, has no solution.} Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing $uv$, $n$ the order of $\frac{v}{u}\bmod q$, $ξ$ a primitive $n$th root of unity and $M:=\Q(ξ,ζ)$. The aim of this complement of the article [GQ…
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields…
Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As an application to non-Archimedian Diophantine approximation in positive characteristic, we relate the growth of the orbits of lattices under one-parameter unipotent subgroups of $\GL_2(\wh K)$ with approximation exponents and continued fraction expansions of elements of the fiel…
Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Padé approximations of the second kind.
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach…
The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law f…
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.
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。
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…
On a Diophantine Equation Involving Lucas Numbers
本文研究了涉及卢卡斯数的丢番图方程，证明了在特定条件下该方程无解，采用了因式分解、数论引理和计算机辅助计算等方法。
Gaussian periods are cyclotomic integers with a long history in number theory and connections to problems in combinatorics. We investigate the asymptotic behavior of the absolute norm of a Gaussian period and provide a rate of convergence in a case of Myerson's Conjecture for periods of arbitrary odd length. Our method involves a result of Bombieri, Masser, and Zannier on unlikely intersections in the algebraic torus as well as work of the author on the diophantine approximations to a set definable in an o-minimal structure. In the appendix we make a result of Lawton on Mahler measures quantit…
On a Diophantine Equation with Jacobsthal and Fibonacci Numbers
研究含雅各布斯塔尔数和斐波那契数的丢番图方程，核心方法是对数线性形式技术，主要贡献是确定\(2<k<l<m\)时，丢番图方程\(F_kF_lF_m = J_n\)唯一解为\((k,l,m,n)=(5,7,8,12)\)。
Effective equidistribution for multiplicative Diophantine approximation on lines
本文通过证明 SL(3,R)/SL(3,Z) 中幂幺轨道的有效等分布，将 Littlewood 猜想在直线上几乎处处加强了两个对数因子，并利用对偶 Bohr 集理论改进 Kleinbock 的收敛结果及齐性空间对数律。
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。
An asymptotic on the logarithms of the relative class numbers of imaginary abelian number fields of prime conductor and large degree
本文研究了具有素数导子 $p$ 和大次数的虚阿贝尔数域，给出了其相对类数对数的渐近性质，并证明了在稍放宽限制条件下该渐近性质不再成立。
Let $ζ_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(ζ_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a ne…
In this paper, we consider the Diophantine equation $λ_1U_{n_1}+\ldots+λ_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; $w$ is a fixed non-zero integer; $p_1,\dots,p_s$ are fixed, distinct prime numbers; $λ_1,\dots,λ_k$ are strictly positive integers; and $n_1,\dots,n_k,z_1,\dots,z_s$ are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $(n_1,\dots,n_k,z_1,\dots,z_s)$. In our proof, we use lower bounds for linear forms in log…
Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above $2$. When modularity of elliptic curves over $K$ is known, for example when $K$ is real quadratic or the $r$-layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, effective asymptoti…
Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all three types of the recursion admit an explicit solution in the form of complete Bell polynomial, depending on the coefficients of the power series expansion of the logarithm of the generating functions for the sequences of individual terms in the Diophantine equations.
The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation turns out to be a difficult task. We focus our attention to $p=5$, by reason of an article by F. Abu Muriefah, published in this journal, vol. 128 (2008), 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equa…
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of `local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in \cite{memoirs}. Fur…
We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmi…
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…
On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$
本文研究丢番图方程 $F_n = F_l^k (F_l^m-1)$，利用 Matveev 定理、改进的 Baker-Davenport 约化方法及斐波那契数整除性质，证明唯一正整数解为 $(6,3,3,1)$。
On the Diophantine Equations $J_n +J_m =L_k$ and $L_n +L_m =J_k$
本文利用 Baker 对数线性形式定理及 Dujella 和 Pethő 的缩减方法，求出了丢番图方程 $L_n +L_m =J_k$ 和 $J_n +J_m =L_k$ 的所有非负整数解，找出了所有可表示为两个 Jacobsthal 数之和的 Lucas 数及反之。
Let $α$ be an irrational real number. We show that the set of $ε$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (α) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| qα-x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $ε$ if and only if $α$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $α$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a part…
The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation
本文研究 Hadamard 空隙序列膨胀集模 1 的最大间隙，证明存在膨胀因子使间隙阶至多为 (log N)/N，且对几乎所有的膨胀因子该阶至多为 (log N)^{2+ε}/N，并由此改进乘法丢番图逼近中 Littlewood 问题非齐次版本的上界。
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。
Mahler's work on Diophantine equations and subsequent developments
本文讨论Mahler在丢番图逼近及其在丢番图方程中应用的工作，并深入探讨Thue-Mahler方程解的个数与有效有限性等后续发展。
Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$ with some conditions on $n_i $ for all $1\leq i \leq t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due t…
Extreme values and logarithm laws for toral translations
该研究针对非齐次丢番图逼近中归一化第r近邻距离函数，基于齐次动力学，结合泊松近似策略与多重等分布结果，证明其极值极限定律及对应对数定律。
On the resolution of the Diophantine equation $U_n + U_m = x^q$
本文研究二元递归序列下Diophantine方程$U_n + U_m = x^q$的解，利用对数线性型证明固定$x$时解有限，并在abc-conjecture下结合Schmidt子空间定理等证明$q,x\geq2$时解有限。
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。
Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $δ(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lovász local lemma to es…
In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.
We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.
On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers
本文研究了Fibonacci和Lucas数的混合b-拼接形式，通过求解两个丢番图方程确定哪些Lucas数可表示为Fibonacci和Lucas数的拼接，证明仅存在有限多个这样的Lucas数。
We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size B with each component having no prime divisors below $B^{1/u}$, where $u=c_0n^{3/2}$, $n$ is the number of variables and $c_0$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $$n^{3/2}$$ to the logarithmic $$\frac{\log n}{\log \log n}.$$ Our main new ingredients are the generalisation of the Brüdern-Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenpor…
Random walks on the circle and Diophantine approximation
本文研究圆周群上的随机游走，将有限状态空间Markov链的中心极限定理推广至无理跨度情形，并揭示了Kolmogorov度量下收敛速率从多项式到指数衰减的新现象。
For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside S. This paper treats the case where S does not contain p and k is the rational number field or an imaginary quadratic field. In this case, we prove the explicit formulae for the free ranks of the S-ramified Iwasawa modules as abelian pro-p groups, by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of a…
Narayana's cows sequence satisfies the third-order linear recurrence relation $N_n=N_{n-1}+N_{n-3}$ for $n \geq 3$ with initial conditions $N_0=0$ and $N_1=N_2=1$. In this paper, we study $b$-repdigits which are sums of two Narayana numbers. We explicitly determine these numbers for the bases $2\le b\leq100$ as an illustration. We also obtain results on the existence of Mersenne prime numbers, 10-repdigits, and numbers with distinct blocks of digits in the Narayana sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms and a version of the Baker-Davenport redu…
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。
Perfect powers in sequences of polygonal numbers
研究多边形数序列中完美幂的丢番图方程，利用模方法、超几何方法、对数线性形式和大量计算，扩展了已知解的范围。
On qualitative aspects of the quantitative subspace theorem
本文基于参数化子空间定理，通过引入线性截面和对数扭曲高度函数等工具，推导了数域有限扩张下大线性系统的丢番图算术不等式，并建立了丢番图例外集的新颖线性散射型结果。
This text is a mostly self-contained textbook-like introduction to classical and modern Iwasawa Theory in Spanish language. It grew out of notes for a course that the authors gave at the annual conference of the Mexican Mathematical Society (SMM) in 2018. It covers some preliminaries from algebraic number theory and profinite algebra, the theory of the Iwasawa algebra and its modules, results about class groups in $\mathbb{Z}_p$-extensions, $p$-adic $L$-functions for Dirichlet characters and their various constructions (along with some complex theory of $L$-functions), the cyclotomic main conj…
On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent
本文研究了完全p-次虚二次域的Z_p扩张性质，不依赖Iwasawa理论，探讨了p-类群的滤子及p-类群的阶，提出新的视角和猜想。
Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and th…
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