Results 21 to 30 of about 93 (91)
Simultaneous diophantine approximation and IP-sets [PDF]
Acta Arithmetica, 1988 A sequence \(p_1, p_2,\ldots\) in \(\mathbb{Z}\) together with all sums \(p_{i_1}+\cdots +p_{i_k ...
Furstenberg, H., Weiss, B.
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ON THE LITTLEWOOD CONJECTURE IN SIMULTANEOUS DIOPHANTINE APPROXIMATION [PDF]
Journal of the London Mathematical Society, 2006 For any given real number $ $ with bounded partial quotients, we construct explicitly continuum many real numbers $ $ with bounded partial quotients for which the pair $( , )$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.
Adamczewski, Boris, Bugeaud, Yann
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Simultaneous diophantine approximation in non-degenerate p-adic manifolds [PDF]
Israel Journal of Mathematics, 2011 S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.
Mohammadi, A., Salehi Golsefidy, A.
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Simultaneous Diophantine approximation of rationals by rationals
Journal of Number Theory, 1986 For positive integers \(n\geq 2\), \(B\geq 2\), let \(S_ n(B)\) denote the set of rational vectors \(\alpha =(a_ 1/B,...,a_ n/B)\) with \(a_ j\in {\mathbb{Z}}\), \(0\leq a_ j0\), define N(\(\alpha\),\(\Delta)\) as the number of vectors \(\zeta =(x_ 1/x,...,x_ n/x)\) with \(1\leq ...
Lagarias, Jeffrey C, Hastad, Johan T
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Strong characterizing sequences in simultaneous diophantine approximation
Journal of Number Theory, 2003 In this paper, it is proved that: if \(1,\alpha_1,\dots, \alpha_t\in\mathbb{R}\) are linearly independent over the rationals, there is a subset \(A\subset\mathbb{N}\), \(| A|=\infty\), such that \(\sum_{n\in A}\| n\beta\|\) is finite if and only if \(\beta\in G\), the group generated by \(1,\alpha_1,\dots, \alpha_t\).
Biró, András, T. Sós, Vera
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A note on simultaneous diophantine approximation
Manuscripta Mathematica, 1981 Refining earlier investigations due to J.M.MACK [7] by a method of MORDELL it is proved that for any two irrational numbers α, β there exist infinitely many pairs of fractions p/r, q/r satisfying the inequalities $$|\alpha - \frac{p}{r}|< \frac{8}{{13}}r^{ - 3/2} ,|\beta - \frac{q}{r}|< \frac{8}{{13}}r^{ - 3/2} .$$ .
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Simultaneous Diophantine approximation - logarithmic improvements
, 2016 This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an ...
Gorodnik, Alexander, Vishe, Pankaj
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Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds
Journal of Number Theory, 1996 Let \(\psi(q) \in \mathbb{N}\) for \(q=1, 2, 3, \dots\) be decreasing such that for some \(k \in \mathbb{N}\) the series \(\sum\psi(q)^k\) is divergent. As a slight variation of Khintchine's theorem on simultaneous diophantine approximation, it is known that, for almost all \((x_1, \dots, x_k) \in\mathbb{R}^k\) there are infinitely many solutions of ...
Dodson, M.M. +2 more
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Topological triple phase transition in non-Hermitian Floquet quasicrystals. [PDF]
Nature, 2022 Weidemann S +3 more
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Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators [PDF]
Transactions of the American Mathematical Society, 1982 This paper defines the notion of a best simultaneous Diophantine approximation to a vector α \alpha in R n R^n with respect to a norm ‖ ⋅ ‖ \left \| \,\cdot \, \right \| on R n R^n .
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