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We consider some families of binary binomial forms $aX^d+bY^d$, with $a$ and $b$ integers. Under suitable assumptions, we prove that every rational integer $m$ with $|m|\ge 2$ is only represented by a finite number of the forms of this family (with varying $d,a,b$). Furthermore {the number of such forms of degree $\ge d_0$ representing $m$ is bounded by $O(|m|^{(1/d_0)+\epsilon})$} uniformly for $\vert m \vert \geq 2$. We also prove that the integers in the interval $[-N,N]$ represented by one of the form of the family with degree $d\geq d_0$ are almost all represented by some form of the family with degree $d=d_0$. In a previous {paper} we investigated the particular case where the binary binomial forms are positive definite. We now treat the general case by using a lower bound for linear forms of logarithms.

According to Lidstone interpolation theory, an entire function of exponential type $<\pi$ is determined by it derivatives of even order at $0$ and $1$. This theory can be generalized to several variables. Here we survey the theory for a single variable. Complete proofs are given. This first paper of a trilogy is devoted to Univariate Lidstone interpolation; Bivariate and Multivariate Lidstone interpolation will be the topic of two forthcoming papers., Comment: 19 pages

We extend our previous results on the number of integers which are values of some cyclotomic form of degree larger than a given value (see \cite{FW1}), to more general families of binary forms with integer coefficients. Our main ingredient is an asymptotic upper bound for the cardinality of the set of values which are common to two non isomorphic binary forms of degree greater than $3$. We apply our results to some typical examples of families of binary forms., Comment: Two misprints corrected - the version to appear in Acta Arithmetica is correct

We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a polynomial. The proof in the one dimensional case relies on Lidstone expansion of the function. For $n$ variables, we need $n+1$ points, having the property that the differences of $n$ of them with the remaining one give a basis of ${{\mathbb C}}^n$. The proof is by reduction to the one variable situation., Comment: To appear in Journal of Ramanujan Society of Mathematics and Mathematical Sciences (JRSMAMS), Vol. 9, No. 1, December 2021

We present a uniform description of sets of $m$ linear forms in $n$ variables over the field of rational numbers whose computation requires $m(n - 1)$ additions.

An integer--valued function is an entire function which maps the nonnegative integers $\mathbb N$ to the integers. An example is $2^z$. A Hurwitz function is an entire function having all derivatives taking integer values at $0$. An example is ${\mathrm e}^z$. Lower bound for the growth order of such functions have a rich history. Many variants have been considered: for instance, assuming that the first $k$ derivatives at the integers are integers, or assuming that the derivatives at $k$ points are integers. These as well as and many other variants have been considered. We survey some of them., Comment: 24 pages

Given a subset $S=\{s_0, s_1\}$ of the complex plane with two points and an infinite subset ${\mathscr S}$ of $S\times {\mathbb N}$, where ${\mathbb N}=\{0,1,2,\dots\}$ is the set of nonnegative integers, we ask for a lower bound for the order of growth of a transcendental entire function $f$ such that $f^{(n)}(s)\in{\mathbb Z}$ for all $(s,n)\in{\mathscr S}$. We first take ${\mathscr S}=\{s_0,s_1\}\times 2{\mathbb N}$, where $2{\mathbb N}=\{0,2,4,\dots\}$ is the set of nonnegative even integers. We prove that an entire function $f$ of sufficiently small exponential type such that $f^{(2n)}(s_0)\in{\mathbb Z}$ and $f^{(2n)}( s_1)\in{\mathbb Z}$ for all sufficiently large $n$ must be a polynomial. The estimate we reach is optimal, as we show by constructing a noncountable set of examples. The main tool, both for the proof of the estimate and for the construction of examples, is Lidstone polynomials. Our second example is $(\{s_0\}\times (2{\mathbb N}+1))\cup( \{ s_1\}\times 2{\mathbb N})$ (odd derivatives at $s_0$ and even derivatives at $ s_1$). We use analogs of Lidstone polynomials which have been introduced by J.M.~Whittaker and studied by I.J.~Schoenberg. Finally, using results of W.~Gontcharoff, A. J.~Macintyre and J.M.~Whittaker, we prove lower bounds for the exponential type of a transcendental entire function $f$ such that, for each sufficiently large $n$, one at least of the two numbers $f^{(n)}(s_0)$, $f^{(n)}(s_1)$ is in ${\mathbb Z}$., Comment: 28 pages

Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$ rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies $f^{(mn+r_j)}(s_j)\in{\mathbb Z}$ for $0\le j\le m-1$ and all sufficiently large $n$, then $f$ is a polynomial. Under suitable assumptions on $s_0,s_1,\dots,s_{m-1}$ and $r_0,\dots,r_{m-1}$, we introduce interpolation polynomials $\Lambda_{nj}$, ($n\ge 0$, $0\le j\le m-1$) satisfying $$ \Lambda_{nj}^{(mk+r_\ell)}(s_\ell)=\delta_{j\ell}\delta_{nk}, \quad\hbox{for}\quad n, k\ge 0 \quad\hbox{and}\quad 0\le j, \ell\le m-1 $$ and we show that any entire function $f$ of sufficiently small exponential type has a convergent expansion $$ f(z)=\sum_{n\ge 0} \sum_{j=0}^{m-1}f^{(mn+r_j)}(s_j)\Lambda_{nj}(z). $$ The case $r_j=j$ for $0\le j\le m-1$ involves successive derivatives $f^{(n)}(w_n)$ of $f$ evaluated at points of a periodic sequence ${\mathbf{w}}=(w_n)_{n\ge 0}$ of complex numbers, where $w_{mh+j}=s_j$ ($h\ge 0$, $0\le j\le m$). More generally, given a bounded (not necessarily periodic) sequence ${\mathbf{w}}=(w_n)_{n\ge 0}$ of complex numbers, we consider similar interpolation formulae $$ f(z)=\sum_{n\ge 0}f^{(n)}(w_n)\Omega_{{\mathbf{w}},n}(z) $$ involving polynomials $\Omega_{{\mathbf{w}},n}(z)$ which were introduced by W.~Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis $f^{(n)}(w_n)\in{\mathbb Z}$ for all sufficiently large $n$ implies that $f$ is a polynomial., Comment: 21 pages. To appear in the Moscow Journal of Combinatorics and Number Theory. The first version was substantially improved thanks to a contribution by Damien Roy

For each integer $d\ge 4$, we study the sequence of positive integers which are represented by one at least of the cyclotomic binary forms $\Phi_n(X,Y)$, with $n$ a positive integer satisfying $\varphi(n)\ge d$. The case $d=2$ was studied in our previous work [FLW]. Our demonstration is based on a variant of a statement of [SX] concerning the common values taken by two binary forms of the same degree and non-zero discriminants. All constants are effectively calculable., Comment: 25 pages. En fran\c{c}ais

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb Z}$, which we write as $$ X^d-U_1(a)X^{d-1}Y+\cdots+(-1)^{d-1} U_{d-1}(a) XY^{d-1}+(-1)^d U_d(a) Y^d.$$ In this paper we study these sequences $\bigl(U_h(a)\bigr)_{a\in {\mathbb Z}}$ which turn out to be linear recurrence sequences., Comment: Proceedings of the International Conference on Pure and Applied Mathematics ICPAM-GOROKA 2014, "Contemporary Developments in the Mathematical Sciences as Tools for Scientific and Technological Transformation of Papua New Guinea"

The homogeneous form $\Phi_n(X,Y)$ of degree $\varphi(n)$ which is associated with the cyclotomic polynomial $\phi_n(X)$ is dubbed a {\it cyclotomic binary form}. A positive integer $m\ge 1$ is said to be {\it representable by a cyclotomic binary form} if there exist integers $n,x,y$ with $n\ge 3$ and $\max\{|x|, |y|\}\ge 2$ such that $\Phi_n(x,y)=m$. We prove that the number $a_m$ of such representations of $m$ by a cyclotomic binary form is finite. More precisely, we have $\,\varphi(n) \le ({2}/ {\log 3})\log m\, $ and $\, \max\{|x|,|y|\} \le ({2}/{\sqrt{3}})\, m^{1/\varphi(n)}.\,$ We give a description of the asymptotic cardinality of the set of values taken by the forms for $n\geq 3$. This will imply that the set of integers $m$ such that $a_m\neq 0$ has natural density 0. We will deduce that the average value of the integers $a_m$ among the nonzero values of $a_m$ grows like $\sqrt{\log \, m}$., Comment: Acta Arithmetica, to appear

Parametric geometry of numbers is a new theory, recently created by Schmidt and Summerer, which unifies and simplifies many aspects of classical Diophantine approximations, providing a handle on problems which previously seemed out of reach. Our goal is to transpose this theory to fields of rational functions in one variable and to analyze in that context the problem of simultaneous approximation to exponential functions., Comment: 20 pages, the proof of a central lemma is simplified and the main construction is shown to be universal. This paper will appear in Mathematika

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $\sigma_1,\sigma_2,\dots,\sigma_d$ the embeddings of $K$ into $\mathbb{C}$, $\alpha$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-\sigma_i(\alpha) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-\sigma_i(\alpha\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effective upper bound for the solutions $(x,y,a)\in{\mathbb{Z}}^3$ of the Diophantine inequalities $$ 0<|F_a(x,y)|\le m $$ for which $xy\not=0$ and ${\mathbb{Q}}(\alpha \upsilon^a)=K$. Our estimate involves an effectively computable constant depending only on $d$; it is explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$, and in terms of the regulator of the number field $K$., Comment: Submitted. 22 pages

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(\lambda)$ of D. Shanks, $\lambda$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1) $ of $\lambda$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters $n$ and $a$., Comment: Expanded version (31p) of a paper to appear in the Journal de Th\'eorie des Nombres de Bordeaux

Let $F$ be an irreducible binary form attached to a number field $K$ of degree $\geq 3$. Let $\epsilon\not\in \{-1, 1\}$ be a totally real unit of $K$. By twisting $F$ with the powers $\epsilon^a$ of $\epsilon$, ($a\in{\mathbf Z}$), we obtain an infinite family $F_a$ of binary forms. Let $m\in{\mathbf Z}$. We give an effective bound for $\max\{|a|, \log|x|, \log|y|\}$ when $a,x,y$ are rational integers satisfying $F_a(x,y)=m$ with $xy\not=0$., Comment: in French, CRM collection "Contemporary Mathematics", AMS, to appear

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider the irreducible polynomial $f_\epsilon(X)\in{\mathbf Z}[X]$ such that $f_\epsilon(\alpha\epsilon)=0$. Let $F_\epsilon(X,Y)\ = Y^df_\epsilon(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,\epsilon)$ of the diophantine equation $|F_\epsilon(x,y)|\leq m$., Comment: Proceedings of the International Meeting on Number Theory HRI 2011, in honor of R. Balasubramanian. To appear

To each non totally real cubic extension $K$ of $\Q$ and to each generator $\alpha$ of the cubic field $K$, we attach a family of cubic Thue equations, indexed by the units of $K$, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.

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.

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X) \in \Z[X]$ such that $f_\varepsilon(\alpha\varepsilon)=0$. Let $F_\varepsilon(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\varepsilon}(X) $ under the condition $F_{\varepsilon}(X,1)=f_{\varepsilon}(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\varepsilon)$ of the diophantine inequation $|F_\varepsilon(x,y)|\le m$. We achieve this goal by restricting ourselves to a subset of units $\varepsilon$ which we prove to be sufficiently large as soon as the degree of $K$ is $\geq 4$., Comment: Moscow Journal of Combinatorics and Number Theory, to appear

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 $\P^1(K)$ minus $3$ points. Finally we extend our study to higher dimensional spaces in connection with Schmidt's Subspace Theorem.

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing the archimedean places and let $\mu$, $\alpha_1,\alpha_2,\alpha_3$ be non--zero elements in $K$. Denote by $\OS$ the ring of $S$--integers in $K$ and by $\OS^\times$ the group of $S$--units. Then the set of equivalence classes (namely, up to multiplication by $S$--units) of the solutions $(x,y,z,\varepsilon_1, \varepsilon_2,\varepsilon_3,\varepsilon)\in\OS^3\times(\OS^\times)^4$ of the diophantine equation $$ (X-\alpha_1 E_1 Y) (X-\alpha_2E_2 Y) (X-\alpha_3E_3 Y)Z=\mu E, $$ satisfying $\Card\{\alpha_1\varepsilon_1,\alpha_2\varepsilon_2,\alpha_3\varepsilon_3\}= 3$, is finite. With the help of this last result, we exhibit new families of Thue-Mahler equations having only trivial solutions. Furthermore, we produce an effective upper bound for the number of these solutions. The proofs of this paper rest heavily on Schmidt's subspace theorem.

In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bella\"iche asked the same question for the special case of Abelian varieties. These problems are $p$--adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual $p$--adic rank is at least the third of the expected value.

After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems., Comment: Submitted for publication in Pillai's collected papers

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. There is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of $x$. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results.

We discuss the role of auxiliary functions in the development of transcendental number theory.

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these works., Comment: to be published by Springer Verlag, Special volume in honor of Serge Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and John Tate

Les nombres r\'eels ou complexes forment un ensemble ayant la puissance du continu. Parmi eux, ceux qui sont int\'eressants, qui apparaissent naturellement, qui m\'eritent notre attention, forment un ensemble d\'enombrable. Dans cet \'etat d'esprit nous nous int\'eressons aux p\'eriodes au sens de Kontsevich et Zagier. Nous faisons le point sur l'\'etat de nos connaissances concernant la nature arithm\'etique de ces nombres: d\'ecider si une p\'eriode est un nombre rationnel, alg\'ebrique irrationnel ou au contraire transcendant est l'objet de quelques th\'eor\`emes et de beaucoup de conjectures. Nous pr\'ecisons aussi ce qui est connu sur l'approximation diophantienne de tels nombres, par des nombres rationnels ou alg\'ebriques., Comment: Proceedings of the 11th symposium of the Tunisian Mathematical Society held in Tunis, Tunisia -- March 15-18, 2004. http://www.african-j-math.org/ http://www.math.jussieu.fr/~miw/articles/pdf/MahdiaPeriodes.pdf

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references., Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartier

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $\xi$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $\xi$ that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal., Comment: The section 4 of this new version has been rewritten to simplify the proof of the main result. Other results in Sections 9 and 10 have been improved. To appear in Compositio Math

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., Comment: 20 pages. To appear in Publ. Math. Debrecen, 56, 3-4 (2000) See also http://www.math.jussieu.fr/~miw/articles/Debrecen.html