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Abstract  Let  Q  be an infinite set of positive integers. Denote by  W   τ , n  ( Q ) (respectively,  W   τ , n  ) the set of points in dimension  n ≥1 that are simultaneously  τ ‐approximable by infinitely many rationals with denominators in  Q  (respectively, in ℕ * ). When  n ≥2 and  τ >1+1/( n −1), a non‐trivial lower bound for the Hausdorff dimension of the liminf set  W   τ , n   ∖ W   τ , n   ( Q ) is established in the case where the set  Q  satisfies some divisibility properties. The computation of the actual value of this Hausdorff dimension and the one‐dimensional analogue of the problem are also discussed.

A NOTE ON THE HAUSDORFF DIMENSION OF SOME LIMINF SETS APPEARING IN SIMULTANEOUS DIOPHANTINE APPROXIMATION