%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 }