Results 51 to 60 of about 93 (91)

Simultaneous Diophantine Approximation

Canadian Journal of Mathematics, 1950
Summary of results. The principal result of this paper is as follows: given any set of real numbers z1, z2, & , zn and an integer t we can find an integer and a set of integers p1, p2 & , pn such that(0.11).Also, if n = 2, we can, given t, produce numbers z1 and z2 such that(0.12)This supersedes the results of Nils Pipping (Acta Aboensis, vol.
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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

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 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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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 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 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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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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On simultaneous diophantine approximations. Vectors of given diophantine type

Mathematical Notes, 1997
Let \(\psi(y)\) be a real-valued function of a real argument. A positive integer \(p\) is called a simultaneous \(\psi\)-approximation for the numbers \(\alpha_1,\dots,\alpha_s\in \mathbb R\) if \[ \max_{1\leq j\leq s}\| p\alpha_j\|\leq \psi(p)\;(\text{here }\|\alpha\|= \min_{z\in \mathbb Z}| \alpha- z|). \] The numbers \(\alpha_1,\dots, \alpha_s\) are
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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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