# Wiener-Wintner theorem

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Wiener–Wintner ergodic theorem

A strengthening of the pointwise ergodic theorem (cf. also Ergodic theory) announced in [a21] and stating that if is a dynamical system, then given one can find a set of full measure such that for in this set the averages converge for all real numbers . In other words, the set "works" for an uncountable number of . This introduces into ergodic theory the study of general phenomena in which sampling is "good" for an uncountable number of systems. Since [a21], several proofs of the "Wiener–Wintner theorem" have appeared (e.g., see [a11] for a spectral path and [a14] for a path using the notion of disjointness in [a13]).

## Uniform Wiener–Wintner theorem and Kronecker factor.

For an ergodic dynamical system (cf. also Ergodicity), the Kronecker factor of is defined as the closed linear span in of the eigenfunctions of . The orthocomplement of can be characterized by the Wiener–Wintner theorem. More precisely, a function is in if and only if for -a.e. with respect to , This theorem was announced by J. Bourgain [a9]. Other proofs of this result can be found in [a1] and [a15], for instance.

A sequence of scalars is a good universal weight (for the pointwise ergodic theorem) if the averages converge -a.e. for all dynamical systems and all functions . Bourgain's return-time theorem states that given a dynamical system and a function in , then for -a.e. with respect to , the sequence is a good universal weight (see [a8]). By applying this result to the irrational rotations on the one-dimensional torus given by and to the function , one easily obtains the Wiener–Wintner theorem. Another proof of his result can be found in [a10] and [a19]. Previous partial results can be found in [a11].

## Wiener–Wintner return-time theorem and the Conze–Lesigne algebra.

A natural generalization of the return-time theorem is its Wiener–Wintner version, in which averages of the sequence are considered. Such a generalization was obtained in [a7] and one of the tools used to prove it was the Conze–Lesigne algebra. This algebra of functions was discovered by J.P. Conze and E. Lesigne [a12] in their study of the norm convergence of the averages (a1)

for . These averages were introduced by H. Furstenberg. (The functions are in . The -norm convergence of (a1) for is still an open problem (as of 2001).) It is shown in [a7] that the orthocomplement of the Conze–Lesigne factor characterizes those functions for which outside a single null set of independent of or one has -a.e. Several results related to the ones above can be found in [a2], [a3], [a4], [a16], [a18], [a20], [a17], and [a22]. In [a5] it was shown that many dynamical systems have a Wiener–Wintner property, based on the speed of convergence in the uniform Wiener–Wintner theorem; this allows one to derive the results in [a8] and [a9] for such systems in a much simpler way.