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).
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