Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth

We construct finitely generated nil algebras with prescribed growth rate. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow distortion. We then move on to examples of nil algebras and domains with strongly oscillating growth functions and construct primitive algebras for which the Gelfand-Kirillov dimension is strictly sub-additive with respect to tensor products, thus answering a question raised by Krempa-Okninski and Krause-Lenagan.