Let s be a positive integer, 0⩽v⩽1, L any subset of positive integers such that ∑qϵlq−v−ε is divergent but ∑qϵlq−v−ε is convergent for every ε>0. Let λ>1+ν/s and denote by Eλ(L) the set of all real s-tuples (α1,…,αs) satisfying the set of inequalities |qxi|≤q1−λ(i=1,…,s) for an infinite number of qϵL. (|α|) denotes the distance from x to the nearest integer.) It is proved that the Hausdorff dimensions of Eλ(L) is (s+ν)/λ. When L is the set of all positive integers, the result specializes to a well-known theorem of Jarnik (Math. Zeitschrift 33, 1931, 505–543). It also includes some results of Eggleston (Proc. Lond. Math. Soc. Series 2 54, 1951, 42–93) about arithmetical progressions and sets of positive density (ν=1) and geometrical progressions (ν=0).
Description
A generalization of jarnik's theorem on diophantine approximations - ScienceDirect
%0 Journal Article
%1 BOROSH1972193
%A Borosh, I
%A Fraenkel, A.S
%D 1972
%J Indagationes Mathematicae (Proceedings)
%K diophantineapproximation
%N 3
%P 193 - 201
%R https://doi.org/10.1016/1385-7258(72)90055-8
%T A generalization of jarnik's theorem on diophantine approximations
%U http://www.sciencedirect.com/science/article/pii/1385725872900558
%V 75
%X Let s be a positive integer, 0⩽v⩽1, L any subset of positive integers such that ∑qϵlq−v−ε is divergent but ∑qϵlq−v−ε is convergent for every ε>0. Let λ>1+ν/s and denote by Eλ(L) the set of all real s-tuples (α1,…,αs) satisfying the set of inequalities |qxi|≤q1−λ(i=1,…,s) for an infinite number of qϵL. (|α|) denotes the distance from x to the nearest integer.) It is proved that the Hausdorff dimensions of Eλ(L) is (s+ν)/λ. When L is the set of all positive integers, the result specializes to a well-known theorem of Jarnik (Math. Zeitschrift 33, 1931, 505–543). It also includes some results of Eggleston (Proc. Lond. Math. Soc. Series 2 54, 1951, 42–93) about arithmetical progressions and sets of positive density (ν=1) and geometrical progressions (ν=0).
@article{BOROSH1972193,
abstract = {Let s be a positive integer, 0⩽v⩽1, L any subset of positive integers such that ∑qϵlq−v−ε is divergent but ∑qϵlq−v−ε is convergent for every ε>0. Let λ>1+ν/s and denote by Eλ(L) the set of all real s-tuples (α1,…,αs) satisfying the set of inequalities |qxi|≤q1−λ(i=1,…,s) for an infinite number of qϵL. (|α|) denotes the distance from x to the nearest integer.) It is proved that the Hausdorff dimensions of Eλ(L) is (s+ν)/λ. When L is the set of all positive integers, the result specializes to a well-known theorem of Jarnik (Math. Zeitschrift 33, 1931, 505–543). It also includes some results of Eggleston (Proc. Lond. Math. Soc. Series 2 54, 1951, 42–93) about arithmetical progressions and sets of positive density (ν=1) and geometrical progressions (ν=0).},
added-at = {2019-03-04T19:42:29.000+0100},
author = {Borosh, I and Fraenkel, A.S},
biburl = {https://www.bibsonomy.org/bibtex/28c053c1f2ac63a51d9c9efa7af06ecd7/reimannjan},
description = {A generalization of jarnik's theorem on diophantine approximations - ScienceDirect},
doi = {https://doi.org/10.1016/1385-7258(72)90055-8},
interhash = {1f8697e4a3efbe3a8bb48cb7ad19761d},
intrahash = {8c053c1f2ac63a51d9c9efa7af06ecd7},
issn = {1385-7258},
journal = {Indagationes Mathematicae (Proceedings)},
keywords = {diophantineapproximation},
number = 3,
pages = {193 - 201},
timestamp = {2019-03-04T19:42:29.000+0100},
title = {A generalization of jarnik's theorem on diophantine approximations},
url = {http://www.sciencedirect.com/science/article/pii/1385725872900558},
volume = 75,
year = 1972
}