  The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group
theory. They are natural examples of self-similar finitely generated periodic
groups. The author constructed their analogue in case of restricted Lie
algebras of characteristic 2, Shestakov and Zelmanov extended this construction
to an arbitrary positive characteristic. Also, the author constructed a family
of 2-generated restricted Lie algebras of slow polynomial growth with a nil
$p$-mapping.
  Now, we construct a family of so called clover 3-generated restricted Lie
algebras $T(\Xi)$, where a field of positive characteristic is arbitrary and
$\Xi$ an infinite tuple of positive integers. We prove that $1\le
\mathrm{GKdim}T(\Xi)\le3$, moreover, the set of Gelfand-Kirillov dimensions of
clover Lie algebras with constant tuples is dense on $[1,3]$. We construct a
subfamily of non-isomorphic nil restricted Lie algebras $T(\Xi_{q,\kappa})$,
where $q\in\mathbb N$, $\kappa\in\mathbb R^+$, with extremely slow quasi-linear
growth of type:
$\gamma_{T(\Xi_{q,\kappa})}(m)=m\big(\ln^{(q)}\!m\big)^{\kappa+o(1)}$, as
$m\to\infty$.
  The present research is motivated by a construction by Kassabov and Pak of
groups of oscillating growth. As an analogue, we construct nil restricted Lie
algebras of intermediate oscillating growth in another paper. We call them
"Phoenix algebras" because, for infinitely many periods of time, the algebra is
"almost dying" by having a "quasi-linear" growth as above, for infinitely many
$n$ the growth function behaves like $\exp(n/(\ln n)^{\lambda})$, for such
periods the algebra is "resuscitating". The present construction of 3-generated
nil restricted Lie algebras of quasi-linear growth is an important part of that
result, responsible for the lower quasi-linear growth in that construction.
