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In this paper we introduce and explore the notion of rigidity group,associated with a collection of finitely many sequences, and show that thisconcept has many, somewhat surprising characterizations of algebraic, spectral,and unitary nature. Furthermore, we demonstrate that these characterizationscan be employed to obtain various results in the theory of genericLebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiplerecurrence, additive combinatorics, and spectral theory. As a consequence ofone of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, thereis no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some$|a_j|=1$ if and only if there is an increasing sequence of natural numbers$(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq\{1,...,\ell\}$ there is a $\mu$-preserving transformation$T_F:[0,1]\rightarrow[0,1]$ ($\mu$ denotes the Lebesgue measure) such that forany measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}\mu(A\capT_F^{-b_jn_k}B)=\begin{cases} \mu(A\cap B),\,\text{ if }j\in F,\\\mu(A)\mu(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this resulthas a natural extension to a wide class of families of sequences.

Coexistence of mixing and rigid behaviors in ergodic theory