Results 11 to 20 of about 1,187,035 (128)
Simultaneous Diophantine approximation of rationals by rationals [PDF]
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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On a problem in simultaneous Diophantine approximation: Schmidt's conjecture [PDF]
Annals of Mathematics, 2011 For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in
Badziahin, Dzmitry +2 more
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Weighted simultaneous Diophantine approximation in a variety of settings [PDF]
, 2021 This thesis considers weighted simultaneous Diophantine approximation in a variety of settings, including approximation over real manifolds, p-adic manifolds and p-adic coordinate hyperplanes.
Ward, Benjamin
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Best simultaneous diophantine approximations under a constraint on the denominator [PDF]
, 2006 We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable ...
Aliev, Iskander, Gruber, Peter M.
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A note on weighted simultaneous Diophantine approximation on manifolds [PDF]
In this note we present an improvement to a recent result due to Beresnevich, Levesley, and Ward (2021) pertaining to weighted simultaneous Diophantine approximation on ...
D Allen (4236019), B Wang (805040)
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Lower bounds for simultaneous Diophantine approximation constants [PDF]
, 2014 summary:After a brief exposition of the state-of-art of research on the (Euclidean) simultaneous Diophantine approximation constants $\theta _s$, new lower bounds are deduced for $\theta _6$ and $\theta _7$
Nowak, Werner Georg
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Metric Diophantine approximation on homogeneous varieties [PDF]
, 2014 We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khinchin and Jarnik theorems. In full generality our results establish simultaneous Diophantine
Nevo, Amos +3 more
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Simultaneous Diophantine approximation
Duke Mathematical Journal, 1946 The simplest problems of Diophantine approximation relate to the approximation of a single irrational number 0 by rational numbers pjq, and the principal question is how small we can make the error 0 — pjq in relation to q for infinitely many approximations. It is well known that this question can be answered almost completely in terms of the continued
Davenport, H., Mahler, K.
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Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations [PDF]
, 2005 summary:We prove an analogue of the convergence part of Khintchine’s theorem for the simultaneous inhomogeneous Diophantine approximation on the Veronese curve $(x,x^2,\ldots ,x^n)$ with respect to the different valuations.
Bernik, Vasily +7 more
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Simultaneous diophantine approximation [PDF]
Bulletin of the Australian Mathematical Society, 1972 AbstractUsing a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — αr)2 < c, r(q — βr)2 < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals α and β, provided that c > 1/2.6394.
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