We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space `(N), any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for c0(N) or `(N), 1 < p <∞. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

We prove the spaceability of set hypercyclic vectors for shift-like operators. Shift-like operators appear naturally as composition on Lp(X), when underlying space X is dissipative. In process proving main theorem, we provide, among other results independent interest, a characterization weakly mixing dissipative bounded distortion.