A generalization of jarnik's theorem on diophantine approximations
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Let s be a positive integer, 0v1, L any subset of positive integers such that q{lunate}lq -v- is divergent but q{lunate}lq -v- is convergent for every >0. Let >1+/s and denote by E (L) the set of all real s-tuples ( 1 ,..., s ) satisfying the set of inequalities |qx i |q 1- (i=1,...,s) for an infinite number of q{lunate}L. (||) denotes the distance from x to the nearest integer.) It is proved that the Hausdorff dimensions of E (L) is (s+)/. 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 (=1) and geometrical progressions (=0). 1972.