SUMMER SCHOOL
Ergodic theory

  1. Almost sure convergence and bounded entropy by J. Bourgain, Israel J. Math. 63 (1988), no. 1, 79-97.

    The presenter should focus on proving Propositions 1 and 2 and the application contained in Proposition 10. Other applications may be mentioned. Inequalities (8) and (9) that are needed in the argument can be briefly proved or taken for granted (for their proof and more on this topic, the presenter can consult the paper by A. Bellow and R. Jones: A Banach principle for $L\sp \infty$. [ Adv. Math. 120 (1996), no. 1, 155-172]). A good outline of the theory of entropy of a Gaussian process can be found in the expository paper M. Lacey, Bourgain's entropy criteria [Convergence in ergodic theory and probability (Columbus, OH, 1993), 249-261, Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996].

    [presenter: John Workman]

  2. Convergence of Conze-Lesigne averages by B. Host and B. Kra, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 493-509, also available from Kra's webpage

    The main goal here is to prove Theorem 1.1. The proof of Theorem 2.1 can be briefly sketched if time permits. An easy proof for Lemma 2.2 can be found in Section 6 from the paper of H. Furstenberg and B. Weiss A mean ergodic theorem for $(1/N)\sum\sp N\sb {n=1}f(T\sp nx)g(T\sp {n\sp 2}x)$ [ Convergence in ergodic theory and probability (Columbus, OH, 1993), 193-227, Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996]. The presenter must give a short minicourse on group extensions, cocycles and Mackey group, to ease the the argument. These constructions are also described in the paper of H. Furstenberg Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions [ J. Analyse Math. 31 (1977), 204--256]. Section 4.4. should be skipped.

    [Tamara Kucherenko]

  3. Pointwise ergodic theorems for arithmetic sets by J. Bourgain, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5-45.

    Part I. This presentation involves the first four sections of the paper. The speaker should first show how Theorem 1 and Theorem 2 can be deduced from quantitative inequalities in Fourier analysis. This is explained in section 2, and on the top half of page 25 . The next thing is to sketch the proof of Lemma 4.1 (section 4), and this can start by discussing the auxiliary results from section 3.

    [Andy Yingst]

  4. Pointwise ergodic theorems for arithmetic sets by J. Bourgain, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5-45.

    Part II. This presentation involves sections 5 through 8 of the paper. The result in the appendix will not be discussed. The speaker should first present the proofs of Theorem 1 and Theorem 2 for $L^2$ functions, then for the general $L^p$ case. An excellent detailed presentation of the $L^2$ case appears also in the paper of V. Bergelson (see page 133-145) Combinatorial and Diophantine Applications of Ergodic Theory [Handbook of Dynamical Systems, vol. 1B, B. Hasselblatt and A. Katok, eds., Elsevier (2005), pp. 745-841, available also from Bergelson's webpage]

    [Victor Lie]

  5. Entropy of convolutions on the circle by E. Lindenstrauss, D. Meiri and Y. Peres, Ann. of Math. 149 (1999), 871-904, also available from Lindenstrauss' webpage

    Part I. This presentation involves the first four sections of the paper. A brief introduction to this topic must include the relevant definitions and properties that are needed throughout the paper (i.e. entropy and its properties, Hausdorff dimension, etc.) and the relation of the results in this paper with Furstenberg's conjecture and other similar results by Lyons, Johnson, Rudolph and Host (both Section 1.1 and the paper of Lyons cited as a refference offer some good background.) This will be followed by a general overview of the new results. The proof of the Bootstrap Lemma and that of Theorem 1.4. can be explained in detail. The proof of Theorem 1.5. in section 3 and the discussion on the p-adic collision exponent in section 4 will be briefly touched, depending on how much time is left. Any examples/open questions from section 10, which are relevant to this first part of the presentation, may also be mentioned.

    [Rob Rhoades]

  6. Entropy of convolutions on the circle by E. Lindenstrauss, D. Meiri and Y. Peres, Ann. of Math. 149 (1999), 871-904, also available from Lindenstrauss' webpage

    Part II. This presentation involves sections 5 through 9 of the paper. The main concern here is proving Theorem 1.1. (see sections 5, 6 and 7) and the derivation of the Corollary 1.2 and Theorem 1.3. (section 9). The ideas behind Theorem 1.8. may be briefly mentioned, if time permits. Any examples/open questions from section 10, which are relevant to this second part of the presentation, may also be mentioned.

    [Shuanglin Shao]

  7. Pointwise Theorems for Amenable Groups by E. Lindenstrauss, Invent. Math. 146 (2001), no. 2, 259-295, also available from Lindenstrauss' webpage

    The focus here should be on presenting the random selection algorithm of Folner sets for discrete groups in Section 2 (Algorithm A) and the implications on the pointwise ergodic theorem for actions of amenable groups in Section 3. The algorithm for general (nondiscrete) groups may be briefly touched if time permits. The maximal inequality in Theorem 3.2. will be presented. An important fact that is worth being emphasized is that Theorem 3.2 relies on transferring the maximal inequality from the group to the measure space, this being possible due to the amenability of G. Theorem 3.3. does not have to be proved, since the way a maximal inequality together with convergence on a dense class implies convergence for the full class of integrable functions is a recurrent theme in this summer school. Section 4 will be skipped. The relevance of the pointwise ergodic theorem in this paper to actions of amenable groups of exponential growth -like the lamplighter group- must be mentioned. This is discussed in Section 5, none of the proofs there need to be presented though (perhaps sketched idf time permits).

    [Zubin Gautam]

  8. A generalization of Birkhoff's pointwise ergodic theorem by A. Nevo and E. M. Stein, Acta. Math. 173 (1994), 135-154.

    This paper addresses the question of pointwise ergodic theorems for actions of non amenable groups. As a consequence, the methods from the paper of Lindenstrauss (see 8) do not apply here, in particular the balls on which the averages are taken grow too fast to resemble a Folner sequence. The method of transfer from the group to the measure space is also not available in the absence of amenability. The approach to the pointwise ergodic theorem is instead based on spectral theory here. The speaker should present the main ideas behind the proofs of Theorem 1 and Theorem 2.

    [Svetlana Butler]

  9. The ergodic theoretical proof of Szemeredi's Theorem by H. Furstenberg, Y. Katznelson and D. Orstein, Bulletin (New Series) of the American Mathematical Society 7 , no. 3 (1982), 527-552.

    Part I. This paper is a beautiful, concise and easy to digest exposition of the original argument of Furstenberg on the ergodic Szemeredi theorem (see Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions [ J. Analyse Math. 31 (1977), 204-256]). This first part of the presentation will be concerned with the first 5 sections of the paper. The main points that need to be emphasized are as follows. The first one is the relevance of the ergodic theoretic approach to the number theoretic problem; this is about the fact that Theorem II implies Theorem I and it is discussed in the first part of section 1. A second issue is Furstenberg's theorem in the particular cases of weakly mixing and compact systems, which are the two opposite poles of the spectrum. This is discussed in section 2 for some concrete examples of such systems, and then in sections 3 and 4 for general systems (Theorems 3.1. and 4.1 will need some special attention). Third, the main ideas behind Theorem 5.1. will be mentioned. If time permits, the general strategy for the proof of Theorem II will be briefly anticipated, perhaps the fact that weak mixing and compact systems will be replaced with weak mixing/compact extensions (The technical definitions of these latter concepts will follow in the second part of the presentation.)This strategy can be sketched at any point of this first presentation, and it should anyway be detailed to a somewhat larger extent in the second part of the presentation.

    [Richard Oberlin]

  10. The ergodic theoretical proof of Szemeredi's Theorem by H. Furstenberg, Y. Katznelson and D. Orstein, Bulletin (New Series) of the American Mathematical Society 7 , no. 3 (1982), 527-552.

    Part II. The second part of the presentation will cover sections 6 through 10. Like in the case of the first part of the presentation, not all the technical details can be shown, but all the main steps should be emphasized. This includes a brief presentation of factors (section 6), Proposition 7.1, the definitions of relative weak mixing and compact extensions, their connection with weak mixing and compact systems, Theorems 8.4 and and 9.1, and finally Theorem 10.1. This presentation will be somewhat eased by the first one, which will have already exposed the same ideas in a more particular setting.

    [Anne McCarthy]

  11. The primes contain arbitrarily long arithmetic progressions by B. Green and T. Tao, available from ArXiv

    Part I. This presentation will cover sections 1 through 8 of the paper. It should start by mentioning a bit of background (keep in mind, however, that by the time this talk is given, the audience will already have been introduced to some of the ideas behind Furstenberg's proof of Szemeredi's theorem). A brief outline of the proof may be sketched, like in section 2. Various steps of this outline may be reemphasized at various other stages. Then comes the discussion about pseudorandom measures in section 3, including Theorem 3.5. whose proof will then be presented in few stages. In particular, the speaker should try to demistify a bit the linear forms condition and the correlation condition. The presentation will continue with a discussion of Gowers (anti)uniformity and Proposition 5.3. Last, the speaker will emphasize the main ideas involved in the proof of Proposition 8.1 and show how Proposition 8.1. implies Theorem 3.5.

    [Alex Gorodnik]

  12. The primes contain arbitrarily long arithmetic progressions by B. Green and T. Tao, available from ArXiv

    Part II. This presentation will cover sections 9 through the Appendix. This second part is more analytic number theoretical in flavour and is mostly concerned with the construction of the pseudorandom measures to which Theorem 3.5. is applied. The first step is to define these measures explicitely and then to prove Theorem 1.1 assuming Proposition 9.1. It should be emphasized why the von Manglot distribution is not a good choice and the role of the so called "W-trick" in the argument. Once the measures constructed, it remains to verify that they are indeed pseudorandom measures. This is obtained in Lemma 9.7 and Propositions 9.8. and 9.10. These rely on some estimates essentially due to Goldston-Yilderim (Propositions 9.5 and 9.6) whose proofs may be sketched. A good exposition of the main ideas in this paper, in connection with similar ideas in ergodic theory is the paper by B. Kra : The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view. [Available from Kra's webpage]

    [Tim Austin]

  13. Polynomial extensions of van der Waerden's and Szemeredi's theorems by V. Bergelson and A. Leibman, Journal of AMS 9, no. 3 (1996), 725-573, also available from Bergelson's webpage

    The speaker should focus here on presenting the main ideas involved in the proofs of Theorems A and C. The argument for Theorem A shares some of the ideas behind the Furstenberg-Katznelson-Orstein proof of Szemeredi's theorem (see 9 and 10 above), yet the novelty here manifests in two ways: first, the transformations have polynomial powers; second, the result is proved for commuting transformations. A special attention should be devoted to proving Theorem C, which requires topological dynamics (section 1). The technicalities in section 2 can be skipped (a more detailed insight into weakly mixing extentions will be given in the talks 9 and 10). More time should be spent with the proof of Theorem A ( presented in section 3). The most important combinatorial corollary, Theorem B, must be mentioned, however the speaker need not bother with its proof.

    [Mike Johnson]