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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 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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ON THE SIMULTANEOUS DIOPHANTINE APPROXIMATION OF NEW PRODUCTS

Analysis, 2000
Let \(K\) denote \(\mathbb Q\) or \({\mathbb Q}(i)\) and let \(O_K\) be the ring of integers of \(K\). Let \(q\) be an element of \(O_K\) with \(|q|>1\) and let \(a\) and \(\alpha\) be non-zero elemts of \(K\) such that \(\pm\alpha, -a\alpha\neq q^j\) for any positive integer \(j\). Let \[ f(z)=\prod_{j=1}^\infty g(zq^{-j}) \] where \(g(z)=\left(1+az-z^
Bundschuh, Peter, Väänänen, Keijo
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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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The computational complexity of simultaneous Diophantine approximation problems

23rd Annual Symposium on Foundations of Computer Science (sfcs 1982), 1982
Let \(a=(a_ 1/b_ 1,...,a_ d/b_ d)\) be a rational vector. An integer q is called a best simultaneous diophantine approximation denominator (BSAD) of a if \(\{\) \(\{\) qa\(\}\) \(\}\leq \{\{q'a\}\}\) for all q'\(\in [1,q]\), where \(\{\{qa\}\}=\max (\{q_ ia_ i/b_ i\})\) is the distance to a nearest integer vector.
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Simultaneous diophantine approximation

2000
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