We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number
α∈(0,1) a commutative nonassociative algebra
Aα whose codimension sequence
cn(Aα),
n=1,2,… , is polynomially bounded and
limlogncn(Aα)=3+α.
As an application we are able to construct a new example of a variety with an infinite basis of identities.