The Classification of Rational Approximations

Grace, John Hilton

doi:10.1112/plms/s2-17.1.247

Cited by 9 publications

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“…On the other hand, if a rational number p/q satisfies this inequality, then by Fatou's theorem (see [13]- [15]) p/q coincides either with a convergent of θ, or with an intermediate fraction neighbouring a convergent. Moreover, by Legendre's theorem (see [11]- [13]) any rational number p/q satisfying the inequality…”

Section: 5mentioning

confidence: 99%

Geometry of Diophantine exponents

German

2023

Russian Math. Surveys

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Diophantine exponents are some of the simplest quantitative characteristics responsible for the approximation properties of linear subspaces of a Euclidean space. This survey is aimed at describing the current state of the area of Diophantine approximation that studies Diophantine exponents and relations they satisfy. We discuss classical Diophantine exponents arising in the problem of approximating zero with the set of the values of several linear forms at integer points, their analogues in Diophantine approximation with weights, multiplicative Diophantine exponents, and Diophantine exponents of lattices. We pay special attention to the transference principle. Bibliography: 99 titles.

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Section: 5mentioning

confidence: 99%

Geometry of Diophantine exponents

German

2023

Russian Math. Surveys

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“…Keep the notation of Theorem 2.2. An almost forgotten result of Fatou [16] (see Grace [18] for a complete proof) asserts that if the rational number a/b satisfies |ξ − a/b| < 1/b 2 , then there exists an integer n such that…”

Section: Proofs Of Theorems 22 and 24 And Of Corollary 23mentioning

confidence: 99%

Extensions of the Cugiani-Mahler theorem

Bugeaud¹

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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In 1955, Roth established that if ξ is an irrational number such that there are a positive real number ε and infinitely many rational numbers p/q with q ≥ 1 and |ξ − p/q| < q −2−ε , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with ε replaced by a function q → ε(q) that decreases very slowly to zero, provided that the sequence of rational solutions to |ξ − p/q| < q −2−ε(q) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani's Theorem and extend it to simultaneous approximation.

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“…Each c(l, ?i, k) lies between 1 and 2 so Legendre's Theorem does not apply. However, by a theorem of Grace [4] (first stated by Fatou [3]; see also Robinson [9]) any solution of (3) is either a convergent Pjfaj or a secondary convergent (Pj±Pj-i)l^<7,rt:<7,-i)-Again, by Legendre's theorem, only convergents can satisfy the right hand side of (3), so we need only consider secondary convergents less than 9. (In fact, for 9 = 9 n , there are no secondary convergents larger than 9.)…”

Section: Proof That the Constants Are Best Possiblementioning

confidence: 99%

Finitely Many Asymmetric Diophantine Approximations

Eggan

1965

Journal of the London Mathematical Society

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The Classification of Rational Approximations

Cited by 9 publications

References 0 publications

Geometry of Diophantine exponents

Geometry of Diophantine exponents

Extensions of the Cugiani-Mahler theorem

Finitely Many Asymmetric Diophantine Approximations

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