Results 71 to 80 of about 788,130 (111)

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
N G Moshchevitin
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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 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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On Simultaneous Diophantine Approximations

Proceedings of the London Mathematical Society, 1946
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Exponents for three-dimensional simultaneous Diophantine approximations [PDF]

Czechoslovak Mathematical Journal, 2012
summary:Let $\Theta = (\theta _1,\theta _2,\theta _3)\in \mathbb {R}^3$. Suppose that $1,\theta _1,\theta _2,\theta _3$ are linearly independent over $\mathbb {Z}$. For Diophantine exponents $$ \begin {aligned} \alpha (\Theta ) &= \sup \{\gamma >0\colon
Nikolay Moshchevitin
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On Optimal Systems and on the Structure of Simultaneous Diophantine Approximations

Analysis (Germany), 1981
Jurkat, W. B., Kratz, W.
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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

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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