A generalization of jarnik's theorem on diophantine approximations

Abstract

Let s be a positive integer, 0⩽ v⩽1, L any subset of positive integers such that ∑ qϵl q −v−ε is divergent but ∑ qϵl q −v−ε is convergent for every ε>0. Let λ>1+ν/ s and denote by E λ (L) the set of all real s-tuples (α 1,…,α s ) satisfying the set of inequalities | qx i |≤ q 1− 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 E λ ( 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).