On the set of points with absolutely convergent trigonometric series

Let { n k } k ⩾ 1 be a sequence of real numbers. Denote E ( { n k } k ⩾ 1 ) the set of points for which the trigonometric series ∑ k ⩾ 1 sin ( n k x ) converges absolutely. It is shown that if t k : = n k + 1 / n k tends to infinity monotonically, the Hausdorff dimension of E ( { n k } k ⩾ 1 ) is given by the formula 1 − l i m s u p k → ∞ log k log t k ; if not, this dimensional formula may be false. This strengthens a former work of Erdös and Taylor.