Let E(ℤ) = {einx}n∈ℤ denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L2(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L2 on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.