A central limit theorem for trigonometric series with bounded gaps

摘要

In this paper it is proved that there exists a sequence {n_k} of integers with 1 ≤ n_(k+1) - n_k ≤ 5 such that the distribution of (cos 2Πn_1x +...+cos2Πn_N)/√N on ([ 0, 1 ], B, dx) converges to a Gaussian distribution. It gives an affirmative answer to the long standing problem on lacunary trigonometric series which ask the existence of series with bounded gaps satisfying a central limit theorem.