On the set of points with absolutely convergent trigonometric series

Let {nk}k⩾1{nk}k⩾1 be a sequence of real numbers. Denote E({nk}k⩾1)E({nk}k⩾1) the set of points for which the trigonometric series ∑k⩾1sin(nkx)∑k⩾1sin(nkx) converges absolutely. It is shown that if tk:=nk+1/nktk:=nk+1/nk tends to infinity monotonically, the Hausdorff dimension of E({nk}k⩾1)E({nk}k⩾1) is given by the formula 1−limsupk→∞logklogtk; if not, this dimensional formula may be false. This strengthens a former work of Erdös and Taylor.