Let $Γ\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $α_1,\ldots,α_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let $ε>0$ be a given real number. One of the main results that we prove in this article is as follows; There exist only finitely many tuples $(u, q, p_1,\ldots,p_r)\inΓ\times\mathbb{Z}^{r+1}$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ for some integer $d\geq 1$ satisfying $|α_i q u|>1$, $α_i q u$ is not a pseudo-Pisot number for some integer $i\in\{1, \ldots, r\}$ and $$ 0<|α_j qu-p_j|<\frac{1}{H^ε(u)|q|^{\frac{d}{r}+\varepsilon}} $$ for all integers $j = 1, 2,\ldots, r$, where $H(u)$ is the absolute Weil height. In particular, when $r =1$, this result was proved by Corvaja and Zannier in [3]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Hančl, Kolouch, Pulcerová and Štěpnička in [4]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.
Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by all but one conjugates of algebraic integers of degree n+1.
Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb R$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $\mathbb Q_p$. We focus on the dimension two and study the quality of the simultaneous approximation to two $p$-adic numbers provided by $p$-adic MCFs, where $p$ is an odd prime. Moreover, given algebraical…
We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers xi_1,...,xi_t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers xi_1,...,xi_t generate over Q a field of transcendence degree one. We provide sharper estimates for example when xi_1,...,xi_t form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence crite…
In this paper, we present a result on using algebraic conjugates to form a sequence of approximations to an algebraic number, and in this way obtain effective irrationality measures for related algebraic numbers. From this result, we are able to generalise Thue's Fundamentaltheorem.
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.
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.
This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and …
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.
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number ξby algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to ξand ξ^2 by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers ξ. We give several properties of these numbers including measures of approximation by rational numbers, by quadratic real numbers and by…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the matrix of the regular representation converge to specific algebraic irrationalities. As an interesting special case, we focus on cubic irrationalities giving a generalization of the Khovanskii matrices for approximating cubic irrationalities. We discuss the quality of such approximations considering both rate of convergence and size of denominators. Moreover, we …
It has been conjectured for some time that, for any integer n\ge 2, any real number ε>0 and any transcendental real number ξ, there would exist infinitely many algebraic integers αof degree at most n with the property that |ξ-α| < H(α)^{-n+ε}, where H(α) denotes the height of α. Although this is true for n=2, we show here that, for n=3, the optimal exponent of approximation is not 3 but (3+\sqrt{5})/2 = 2.618...
In this note, we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on manifolds.
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.
For $θ$ a non-algebraic point on a quasi projective variety over a number field, I prove that $θ$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields…
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.
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).
In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.
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 $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ 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 $ξ$ that is either transcendental or algebraic of sufficiently large degree…
In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Padé approximations of the functions under consideration, then apply the functional equations to get an infinite sequence of approximations and use the numerical approximations obtained from this sequence.
Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.
We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever they make sense in the various completions (Archimedean or $p$-adic) of a number field containing these algebraic numbers.
In 1969, H. Davenport and W. Schmidt established a measure of simultaneous approximation for a real number ξand its square by rational numbers with the same denominator, assuming only that ξis not rational nor quadratic over Q. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three.
In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.
About fifty years ago Mahler proved that if $α>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $α^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $α$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $α$ such that the continued fraction of $α^n$ has period length tending to infinity.
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.
This is an expanded version of my talk given at the workshop "Hot Topics: Thin Groups and Super-strong Approximation" (MSRI, Berkeley, February 6-10, 2012).
We give an algorithm that finds a sequence of approximations with Dirichlet coefficients bounded by a constant only depending on the dimension. The algorithm uses the LLL-algorithm for lattice basis reduction. We present a version of the algorithm that runs in polynomial time of the input.
The goal of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to $p$-adic numbers. Firstly, we establish complete analogues of Khintchine's theorem, the Duffin-Schaeffer theorem and the Jarník-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. Th…
We solve the convergence case of the generalized Baker-Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the approximation function. In order to prove these results, we establish asymptotic formulae for the number of rational points close to an affine subspace. One key ingredient is a sharp upper bound on a certain sum of reciprocals of fractional parts associated with the matrix defining the affine subspace.
There are abundant results on Diophantine approximation over fields of positive characteristic (see the survey papers [13, 25]), but there is very little information about simultaneous approximation. In this paper, we develop a technique of \geometry of numbers" in positive characteristic, so that we may generalize some of the classical results on simultaneous approximation to the case of function fields. More precisely, we approximate a finite set of Laurent series by rational functions with a common denominator. In particular, the lower bound results we obtain may be regarded as a high dimen…
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…
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.
We show that Mahler's classification of real numbers $ζ$ with respect to the growth of the sequence $(w_{n}(ζ))_{n\geq 1}$ is equivalently induced by certain natural assumptions on the decay of the sequence $(λ_{n}(ζ))_{n\geq 1}$ concerning simultaneous rational approximation. Thereby we obtain a much clearer picture on simultaneous approximation to successive powers of a real number in general. Another variant of the Mahler classification concerning uniform approximation by algebraic numbers is shown as well. Our method has several applications to classic exponents of Diophantine approximatio…
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.
The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $θ$ to two properly intersecting cycles in projective space with the derivatives of the algebraic distance of $θ$ to their intersection. This improvement can be used to make algebraic independence criteria, to be proved in a forthcoming paper, more flexible and to refine Approximation results that are proved using the metric Bezout Theorem.
We introduce a new exponent of simultaneous rational approximation $\widehatλ_{\min}(ξ,η)$ for pairs of real numbers $ξ,η$, in complement to the classical exponents $λ(ξ,η)$ of best approximation, and $\widehatλ(ξ,η)$ of uniform approximation. It generalizes Fischler's exponent $β_0(ξ)$ in the sense that $\widehatλ_{\min}(ξ,ξ^2) = 1/β_0(ξ)$ whenever $λ(ξ,ξ^2) = 1$. Using parametric geometry of numbers, we provide a complete description of the set of values taken by $(λ,\widehatλ_{\min})$ at pairs $(ξ,η)$ with $1$, $ξ$, $η$ linearly independent over $\mathbf{Q}$.
We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the intensely studied problem of approximation by algebraic numbers (and integers) of bounded degree. We establish the answer to a question of Bugeaud concerning approximation to transcendental real numbers by quadratic irrational numbers, and thereby we refine a result of Davenport and Schmidt from 1967. We also obtain several new characterizations of Liouville number…
We present a hypergeometric construction of rational approximations to $ζ(2)$ and $ζ(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $ζ(2)$ and $ζ(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S. Fischler.
We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of Davenport and Schmidt. As a consequence, we get a sharper lower bound on the exponent of approximation of such a number xi by algebraic integers of degree at most 4.
For any given real number $α$ with bounded partial quotients, we construct explicitly continuum many real numbers $β$ with bounded partial quotients for which the pair $(α, β)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.
It is known that, for each real number x such that 1,x,x^2 are linearly independent over Q, the uniform exponent of simultaneous approximation to (1,x,x^2) by rational numbers is at most (sqrt{5}-1)/2 (approximately 0.618) and that this upper bound is best possible. In this paper, we study the analogous problem for Q-linearly independent triples (1,x,x^3), and show that, for these, the uniform exponent of simultaneous approximation by rational numbers is at most 2(9+sqrt{11})/35 (approximately 0.7038). We also establish general properties of the sequence of minimal points attached to such trip…
Let $ξ, ζ$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $τ$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q ξ\|, \|q ζ\| \} > q^{-1 + τ}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.
The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well approximable points lying on a curve which is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the d…
