We use 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
$\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$
and let
$\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$
be asymptotically linearly independent (that is, for any
$(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$
,
$\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 for any measurable
$A,B\subseteq [0,1]$
and any
$j\in \{1,\ldots ,N\}$
, is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv.29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).