# Wiener-Wintner theorem

*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.

#### References

[a1] | I. Assani, "A Wiener–Wintner property for the helical transform" Ergod. Th. Dynam. Syst. , 12 (1992) pp. 185–194 |

[a2] | I. Assani, "A weighted pointwise ergodic theorem" Ann. IHP , 34 (1998) pp. 139–150 |

[a3] | I. Assani, "Uniform Wiener–Wintner theorems for weakly mixing dynamical systems" Preprint unpublished (1992) |

[a4] | I. Assani, "Strong laws for weighted sums of independent identically distributed random variables" Duke Math. J. , 88 : 2 (1997) pp. 217–246 |

[a5] | I. Assani, "Wiener–Wintner dynamical systems" Preprint (1998) |

[a6] | I. Assani, "Multiple return times theorems for weakly mixing systems" Ann. IHP , 36 : 2 (2000) pp. 153–165 |

[a7] | I. Assani, E. Lesigne, D. Rudolph, "Wiener–Wintner return times ergodic theorem" Israel J. Math. , 92 (1995) pp. 375–395 |

[a8] | J. Bourgain, "Return times sequences of dynamical systems" Preprint IHES (1988) |

[a9] | J. Bourgain, "Double recurrence and almost sure convergence" J. Reine Angew. Math. , 404 (1990) pp. 140–161 |

[a10] | J. Bourgain, H. Furstenberg, Y. Katznelson, D. Ornstein, "Appendix to: J. Bourgain: Pointwise ergodic theorems for arithmetic sets" IHES , 69 (1989) pp. 5–45 |

[a11] | A. Bellow, V. Losert, "The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences" Trans. Amer. Math. Soc. , 288 (1995) pp. 307–345 |

[a12] | J.P. Conze, E. Lesigne, "Théorèmes ergodiques pour des mesures diagonales" Bull. Soc. Math. France , 112 (1984) pp. 143–175 |

[a13] | H. Furstenberg, "Disjointness in ergodic theory" Math. Systems Th. , 1 (1967) pp. 1–49 |

[a14] | E. Lesigne, "Théorèmes ergodiques pour une translation sur une nilvariete" Ergod. Th. Dynam. Syst. , 9 (1989) pp. 115–126 |

[a15] | E. Lesigne, "Spectre quasi-discret et thèoréme ergodique de Wiener–Wintner pour les polynômes" Ergod. Th. Dynam. Syst. , 13 (1993) pp. 767–784 |

[a16] | E. Lesigne, "Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener–Wintner" Ergod. Th. Dynam. Syst. , 10 (1990) pp. 513–521 |

[a17] | D. Ornstein, B. Weiss, "Subsequence ergodic theorems for amenable groups" Israel J. Math. , 79 (1992) pp. 113–127 |

[a18] | E.A. Robinson, "On uniform convergence in the Wiener Wintner theorem" J. London Math. Soc. , 49 (1994) pp. 493–501 |

[a19] | D. Rudolph, "A joinings proof of Bourgain's return times theorem" Ergod. Th. Dynam. Syst. , 14 (1994) pp. 197–203 |

[a20] | D. Rudolph, "Fully generic sequences and a multiple-term return times theorem" Invent. Math. , 131 : 1 (1998) pp. 199–228 |

[a21] | N. Wiener, A. Wintner, "Harmonic analysis and ergodic theory" Amer. J. Math. , 63 (1941) pp. 415–426 |

[a22] | P. Walters, "Topological Wiener–Wintner ergodic theorem and a random ergodic theorem" Ergod. Th. Dynam. Syst. , 16 (1996) pp. 179–206 |

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Wiener–Wintner theorem.

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