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.
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 …
We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by rational points with any prescribed denominator and an almost prime numerator.
Let $F \subseteq [0,1]$ be a set that supports a probability measure $μ$ with the property that $ |\widehatμ(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to $F$ and (ii) the denominators of the `shifted' rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx {\rm \ mod \, } 1)…
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.
We calculate the measure and Hausdorff dimension of sets of matrices over fields of formal power series with good approximation properties for a restricted set of denominators.
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.
We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth numbers tailored to the application. The new exponent approaches 1/3 as the exponent of log n becomes large.
In the thirties of the last century, I. M. Vinogradov proved that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1…
We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_nθ, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $θ$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost…
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.
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.
In this work we proof the following theorem which is, in addition to someother lemmas, our main result:\noindent \textbf{theorem}. Let$\ X=\{ ( x\_{1}\text{, }%t\_{1}) \text{, }( x\_{2}\text{, }t\_{2}) \text{, ..., }(x\_{n}\text{, }t\_{n})\} $ be a finite part of $\mathbb{R}\times \mathbb{R}^{\ast +}$, then there exist a finite part $R$ of $\mathbb{R}%^{\ast +}$ such that for all $\varepsilon > 0$ there exists $r\in R$ such that if $0 < \varepsilon \leq r$ then there exist rational numbers $( \dfrac{p\_{i}}{q}) \_{i=1,2,...,n}$ such that:\{c}| x\_{i}-\dfrac{p\_{i}}{q}| \leq \varepsilon t\_{i} …
We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $pγ$, $p$ running over the primes and $γ$ being a fixed irrational, for Gaussian primes.
We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarnik type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess's bound on short character sums.
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…
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 the…
Let Q be an infinite set of positive integers. Denote by W_{τ, n}(Q) (resp. W_{τ, n}) the set of points in dimension n simultaneously τ--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial lower bound for the Hausdorff dimension of the liminf set W_{τ, n}\W_{τ, n}(Q) is established when n>1 and τ>1+1/(n-1) in the case where the set Q satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension as well as the one--dimensional analogue of the problem are also discussed.
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…
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 apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\R^n$ and their nondegenerate submanifolds.
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…
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.
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.
I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed.
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.
In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two appropriate trivial parametric solutions and obtaining infinitely many nontrivial parametric solutions. Also we work out some examples, in particular the Diophantine systems of A^k+B^k+C^k=D^k+E^4; k=1,3.
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…
The paper is devoted to the problem of estimating the constant of the best Diophantine approximations. The estimates of lower bound $ C_n $ for $ n = 5 $ and $ n = 6 $ was improved. The first chapter gives an overview of the history of estimates of the constant of the best Diophantine approximations. In the second chapter sets out the methods and results of numerical experiments leading to estimates of $ C_n $. The third chapter is devoted to estimating some functions by means of which in the fourth chapter we will proof the estimates of $ C_n $ for $ 3 \leq n \leq 6 $.
We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{π^2}{3} \quad \text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
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.
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 give several upper bounds for the uniform simultaneous Diophantine exponent $\widehatλ_n(ξ)$ of a transcendental number $ξ\in\mathbb{R}$. The most important one relates $\widehatλ_n(ξ)$ and the ordinary simultaneous exponent $ω_k(ξ)$ in the case when $k$ is substantially smaller than $n$. In particular, in the generic case $ω_k(ξ)=k$ with a properly chosen $k$, the upper bound for $\widehatλ_n(ξ)$ becomes as small as $\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of $\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper …
For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.
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…
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examp…
We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector $\underlineξ=(ξ,ξ^{2},\ldots,ξ^{N})$, in various dimensions $N$. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, …
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…
In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/…
