Results 71 to 80 of about 114 (107)

Simultaneous Diophantine Approximation

Proceedings of the London Mathematical Society, 1952
Proof of the theorem: ``Let \(c > 46^{-1/4}\). Then, for every pair of real irrational numbers \(\alpha, \beta\), there exist infinitely many solutions \(p, q, r > 0\) of \(r(p-\alpha r)^2 < c\), \(r(q- \beta r)^2 < c\) in integers.'' This result slightly improves one by \textit{P. Mullender} [Ann. Math. (2) 52, 417-426 (1950; Zbl 0037.17102)].
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Simultaneous asymptotic Diophantine approximations

Mathematika, 1967
Let θ 1 , …, θ k be k real numbers. Suppose ψ( t ) is a positive decreasing function of the positive variable t . Define λ( N ), for all positive integers N , to be the number of solutions in integers p 1 …, p k , q of the inequalities ...
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Simultaneous Diophantine Approximation Using Primes

Bulletin of the London Mathematical Society, 1988
The authors consider k-tuples of reals \((\alpha_ 1,...,\alpha_ k)\) which satisfy the compatibility condition: If \(h_ i\in {\mathbb{Z}}\) for \(1\leq i\leq k\) and \(\sum^{k}_{i=1}h_ i\alpha_ i\in {\mathbb{Q}}\) then \(\sum^{k}_{i=1}h_ i\alpha_ i\in {\mathbb{Z}}\).
Balog, A., Friedlander, J.
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Simultaneous Diophantine Approximation To Series

Journal of the London Mathematical Society, 1959
Es sei \(\mathfrak K\) die Menge aller formalen Laurentreihen \(x = \alpha_d z^d + \alpha_{d-1}z^{d-1}+ \ldots\) mit Koeffizienten aus einem Körper \(\mathfrak k\). Es werde ferner \(\mathfrak T = \mathfrak k [z]\) und \(\mathfrak R = \mathfrak k(z)\) gesetzt.
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SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

Bulletin of the Australian Mathematical Society, 2020
Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions.
WEILIANG WANG, LU LI
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On simultaneous diophantine approximation

Rendiconti del Circolo Matematico di Palermo, 1984
For given \(s\in {\mathbb{N}}\), let \(\theta_ s\) denote the supremum of all reals c with the property that, for any vector \({\bar \alpha}=(\alpha_ 1,...,\alpha_ s)\in({\mathbb{R}}^ s-{\mathbb{Q}}^ s),\) there exist infinitely many \((\bar p,q)\in {\mathbb{Z}}^ s\times {\mathbb{N}}\) satisfying \(| {\bar \alpha}-(1/q)\bar p| \leq c^{-1/s}\quad q^{-1 ...
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An application of simultaneous diophantine approximation in combinatorial optimization

Combinatorica, 1987
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
András Frank, Éva Tardos
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Simultaneous diophantine approximations and Hermite's method

Bulletin of the Australian Mathematical Society, 1980
In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: letnε N* and αl, …, αnbe distinct non-zero rational numbers; there exists a constantc=c(n, αl, …, αn) ≥ 0 such thatfor every non-zero integer point (qo,ql, …,qn)andq= max {|ql|, … |qn|, 3}.
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Simultaneous diophantine approximations with nonmonotonic error function

Doklady Mathematics, 2011
The paper under review contains the announcement of two results along with sketches of their proofs. Both are concerned with algebraic approximation. The first result concerns the inequality \[ | P(x) +d | < \psi(H(P)), \] where \(d\) is a fixed real number, \(P\) varies over the integer polynomials of degree at most \(n \geq 2\) and \(H(P)\) denotes ...
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Hausdorff Dimension and Generalized Simultaneous Diophantine Approximation

Bulletin of the London Mathematical Society, 1998
Suppose that \(m\) is a positive integer, \(\underline{\tau}=(\tau_1,\dots,\tau_m)\) is a vector of positive real numbers, and \(Q\) is an infinite set of positive integers. Let \(W_Q(m;\underline{\tau})\) be the set of points \(\mathbf x=(x_1,\dots,x_m)\in \mathbb R^m\) for which the inequalities \(\|x_iq\|
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