Best Diophantine approximations

Moshchevitin, Nikolay

doi:10.1017/cbo9780511721472.006

Cited by 10 publications

(13 citation statements)

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“…Thereby we recover Moshchevitin's result that R(ξ) = 3 can be reached, i.e. Γ n = ∅, with a new proof, that we consider easier than the original one from [13].…”

Section: On Problemsupporting

confidence: 74%

“…It is well-known that for n ≥ 2 and totally irrational ξ ∈ R n , we have R(ξ) ≥ 3. See [13,Theorem 1.2] for the claim with its proof sketched in the same paper [13, § 1.3], and [14,Theorem 7] for generalizations to a system of linear forms. On the other hand, Moshchevitin [13] showed that the following sets are not empty.…”

Section: Best Approximations In Small Sublatticesmentioning

confidence: 99%

“…An explicit lower bound slightly exceeding n−2+1/(n 2 +1) for the Hausdorff dimension of Θ n can be readily deduced from the proof of Theorem 2.1 below, see formula (62). Inserting small n in this strengthened bound (62), the decimal expansions start with (13) dim…”

Section: On Problemmentioning

confidence: 99%

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Dirichlet spectrum for one linear form

Schleischitz

2023

Bulletin of London Math Soc

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Consider the integer best approximations of a linear form in n ≥ 2 real variables. While it is well-known that any tail of this sequence always spans a lattice of dimension at least three, Moshchevitin showed that this bound is sharp for any n ≥ 2. In this paper, we determine the exact Hausdorff and packing dimension of the set where equality occurs, in terms of n. Moreover, independently we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of Z n+1 . Our lattices jointly span a lattice of dimension three only, thereby leading to an alternative constructive proof of Moshchevitin's result. We determine the packing dimension and up to a small error term O(n −1 ) also the Hausdorff dimension of the according set. Our method combines a new construction for a linear form in two variables n = 2 with a result by Moshchevitin to amplify them. We further employ the recent variatonal principle and some of its consequences, as well as estimates for Hausdorff and packing dimensions of Cartesian products and fibers. Our method permits much freedom for the induced classical exponents of approximation.

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“…Thereby we recover Moshchevitin's result that R(ξ) = 3 can be reached, i.e. Γ n = ∅, with a new proof, that we consider easier than the original one from [13].…”

Section: On Problemsupporting

confidence: 74%

Section: Best Approximations In Small Sublatticesmentioning

confidence: 99%

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Dirichlet spectrum for one linear form

Schleischitz

2023

Bulletin of London Math Soc

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“…For further results on best approximations we refer the reader to the articles [13], [14] and the survey [17].…”

Section: Best Approximationsmentioning

confidence: 99%

Hausdorff dimension of the set of singular pairs

Cheung

2011

Ann. Math.

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In this paper we show that the Hausdorff dimension of the set of singular pairs is . We also show that the action of diag(e t , e t , e −2t ) on SL3R/SL3Z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A. N. Starkov. As a by-product of the analysis, we obtain a higher-dimensional generalization of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the technique used to compute Hausdorff dimension, we reprove a result of I. J. Good asserting that the Hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2.

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“…Отметим, что А. Я. Хинчин в работе [8] приводит простое и короткое дока-зательство равенства В. Ярника (91).…”

Section: теоремы переносаunclassified