Coexistence of mixing and rigid behaviors in ergodic theory

In this paper, we introduce and explore the notion of rigidity group , associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of [ 0 , 1 ] $[0,1]$ left bracket 0 comma 1 right bracket , IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results, we show that given ( b 1 , … , b ℓ ) ∈ N ℓ (b₁,… , b_ )∈ N^ left parenthesis b 1 comma ellipsis comma b Subscript script l Baseline right parenthesis element of double struck upper N Superscript script l , there is no orthogonal vector ( a 1 , … , a ℓ ) ∈ Z ℓ (a₁,… ,a_ )∈ Z^ left parenthesis a 1 comma ellipsis comma a Subscript script l Baseline right parenthesis element of double struck upper Z Superscript script l with some | a j | = 1 |aⱼ|=1 StartAbsoluteValue a Subscript j Baseline EndAbsoluteValue equals 1 if and only if there is an increasing sequence of natural numbers ( n k ) k ∈ N (nₖ)k∈ N left parenthesis n Subscript k Baseline right parenthesis Subscript k element of double struck upper N with the property that for each F ⊆