  We utilize Gaussian measure preserving systems to prove the existence and
genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow
[0,1]$ which exhibit both mixing and rigidity behavior along families of
asymptotically linearly independent sequences. Let
$\lambda_1,...,\lambda_N\in[0,1]$ and let $\phi_1,...,\phi_N:\mathbb
N\rightarrow\mathbb Z$ be asymptotically linearly independent (i.e. for any
$(a_1,...,a_N)\in\mathbb Z^N\setminus\{\vec 0\}$,
$\lim_{k\rightarrow\infty}|\sum_{j=1}^Na_j\phi_j(k)|=\infty$). Then the class
of invertible Lebesgue measure preserving transformations
$T:[0,1]\rightarrow[0,1]$ for which there exists a sequence $(n_k)_{k\in\mathbb
N}$ in $\mathbb N$ with $$\lim_{k\rightarrow\infty}\mu(A\cap T^{-\phi_j(n_k)
}B)= (1-\lambda_j)\mu(A\cap B)+\lambda_j\mu(A)\mu(B),$$ for any measurable
$A,B\subseteq [0,1]$ and any $j\in\{1,...,N\}$, is generic. This result is a
refinement of a result due to A. M. St\"epin (see Theorem 2 in "Spectral
properties of generic dynamical systems") and a generalization of a result due
to V. Bergelson, S. Kasjan, and M. Lema\'nczyk (see Corollary F in "Polynomial
actions of unitary operators and idempotent ultrafilters").
