Spectral synthesis for exponentials and logarithmic length

We study hereditary completeness of systems exponentials on an interval such that the corresponding generating function G is small outside a lacunary sequence intervals Ik. show that, under some technical conditions, exponential system hereditarily complete if and only logarithmic length union these infinite, i.e., $$\sum\nolimits_k {\int_{{I_k}} {{{dx} \over {1 + \left| x \right|}} = \infty } $$ .