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...1, . . . , Tl (or the TiT −1 j ) are individually ergodic. This reduction is standard and performed for instance in [2, page 157], so we only sketch it here. Using the ergodic decomposition (see e.g. =-=[6]-=-) one can disintegrate µ as an integral of measures µy, such that each µy is invariant and ergodic with x)...fl(T n l x) are respect to the Zl action. By hypothesis, the averages En∈[N]f1(T n 1 conver...

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...orrespondence principle in reverse to deduce the above ergodic theory result from a purely combinatorial result (much as the Furstenberg recurrence theorem [5] can be deduced from Szemerédi’s theorem =-=[22]-=-). More precisely, we shall deduce Theorem 1.1 from the following “finitary” version, in which the general measure-preserving system (X, X, µ, T1, . . . , Tl) has been replaced by the finite abelian g...

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... [23], in which functions such as F play a key role in establishing a hypergraph regularity lemma. We shall establish Theorem 1.6 by “finitary ergodic theory” techniques, reminiscent of those used in =-=[8]-=- to establish arbitrarily long arithmetic progressions in the primes. For instance, instead of building infinitary characteristic factors as was done in earlier work on this problem, we shall build fi...

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... which is based on running the Furstenberg correspondence principle in reverse to deduce the above ergodic theory result from a purely combinatorial result (much as the Furstenberg recurrence theorem =-=[5]-=- can be deduced from Szemerédi’s theorem [22]). More precisely, we shall deduce Theorem 1.1 from the following “finitary” version, in which the general measure-preserving system (X, X, µ, T1, . . . , ...

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...hence convergent, in L2 (X, X, µ), as desired. Henceforth we assume the Z l action to be ergodic on (X, X, µ). Since Z l is an amenable group, the Birkhoff pointwise ergodic theorem applies (see e.g. =-=[17]-=-), and in particular we see that for any f ∈ L∞ (X, X, µ) that (3) lim P →∞ Ev∈[P] lf(T v ∫ x0) = f dµ for almost every x0, where we adopt the convention T (v1,...,vl) := T v1 1 X . . . T vl l . Since...

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...rison with Example 1.5 yields (8) in the l = 2 case. Remark 3.20. The above elementary arithmetic manipulations are essentially the same manipluations used in the hypergraph approach (see [21], [19], =-=[10]-=-, [23]) to Szemerédi’s theorem [22] or the Furstenberg-Katznelson theorem [7], in order to rewrite the problem in a “hypergraph” form.12 TERENCE TAO The operator ∆N is clearly linear. For future refe...

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...tal systems. In the special case Ti = T i for some measure-preserving transformation T : X → X, this result of the Ti and the TiT −1 j 12 TERENCE TAO was first obtained for general l by Host and Kra =-=[11]-=- (with a different proof given subsequently by Ziegler [30]). All of the preceding arguments mentioned above approach the norm convergence problem through the techniques of ergodic theory, for instanc...

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...nspired (albeit somewhat indirectly) by the arguments in [2].4 TERENCE TAO “Koopman-von Neumann” counterparts to such regularity lemmas (analogous to the “weak regularity lemma” of Frieze and Kannan =-=[4]-=-). As with other applications of graph and hypergraph methods, the Z l group action in fact plays remarkably little role in these arguments, although the standard fact that this group is amenable 3 wi...

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...ementary arithmetic manipulations are essentially the same manipluations used in the hypergraph approach (see [21], [19], [10], [23]) to Szemerédi’s theorem [22] or the Furstenberg-Katznelson theorem =-=[7]-=-, in order to rewrite the problem in a “hypergraph” form.12 TERENCE TAO The operator ∆N is clearly linear. For future reference we also observe the module identity (9) ∆N(g {1,...,l}h) = g {1,...,l}∆...

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... M ≤ M ∗ such that (10) ‖∆N(g) − ∆N ′(g)‖L2(Z l+1 P for all M ≤ N, N ′ ≤ F(M). ×X) ≤ ε Remark 4.2. This theorem is faintly reminiscent of the “hypergraph counting lemmas” which appear for instance in =-=[16]-=-, [10], [23]. The deduction of Theorem 4.1 from Theorem 1.6 is immediate by specialising to the case where d = l and M∗ = J = 1, where X is a point, and g is the function ∏ l i=1 g {1,...,l+1}\{i}, wh...

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... these averages are Cauchy, hence convergent, in L2 (X, X, µ), as desired. Henceforth we assume the Z l action to be ergodic on (X, X, µ). Applying the Birkhoff pointwise ergodic theorem for Z l (see =-=[28]-=-), and in particular we see that for any f ∈ L ∞ (X, X, µ) that (3) lim P →∞ E v∈[P] lf(T v x0) = for almost every x0, where we adopt the convention T (v1,...,vl) := T v1 1 ∫ X . . . T vl l . f dµ 6 A...

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...preserving transformation T : X → X, this result of the Ti and the TiT −1 j 12 TERENCE TAO was first obtained for general l by Host and Kra [11] (with a different proof given subsequently by Ziegler =-=[30]-=-). All of the preceding arguments mentioned above approach the norm convergence problem through the techniques of ergodic theory, for instance by constructing characteristic factors for the above syst...

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...ive measure is covered by countably many measurable sets, then at least one of those sets also has positive measure. □CONVERGENCE OF MULTIPLE ERGODIC AVERAGES 31 that of the Paris-Harrington theorem =-=[18]-=-. Note that it was established in [29] (see also [20]) that the Lebesgue dominated convergence theorem is equivalent in the reverse mathematics sense to the arithmetic comprehension axiom (ACA), which...

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...cks based on locating several disjoint intervals of the form [M, F(M)], as was recently carried out in [26]. Indeed our arguments here have some of the “multiscale analysis” flavour of [26]. See also =-=[23]-=-, in which functions such as F play a key role in establishing a hypergraph regularity lemma. We shall establish Theorem 1.6 by “finitary ergodic theory” techniques, reminiscent of those used in [8] t...

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...y norm convergence). Let l ≥ 1 be an integer, let F : N → N be a function, and let ε > 0. Then there exists an integer M ∗ > 0 with the following , then property: If P ≥ 1 and f1, . . . , fl : Zl P → =-=[0, 1]-=- are arbitrary functions on ZlP there exists an integer 1 ≤ M ≤ M ∗ such that we have the “L2 metastability” (1) ‖AN(f1, . . .,fl) − AN ′(f1, . . .,fl)‖ L 2 (Z l P ) ≤ ε for all M ≤ N, N ′ ≤ F(M), whe...

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... comparison with Example 1.5 yields (8) in the l = 2 case. Remark 3.20. The above elementary arithmetic manipulations are essentially the same manipluations used in the hypergraph approach (see [21], =-=[19]-=-, [10], [23]) to Szemerédi’s theorem [22] or the Furstenberg-Katznelson theorem [7], in order to rewrite the problem in a “hypergraph” form.12 TERENCE TAO The operator ∆N is clearly linear. For futur...

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... weaker 1 In proof theory, this finitisation is known as the Gödel functional interpretation of the infinitary statement, which is also closely related to the Kriesel no-counterexample interpretation =-=[13]-=-, [14] or Herbrand normal form of such statements; see [12] for further discussion. We thank Ulrich Kohlenbach for pointing out this connection. 2 This is analogous to how the argument in [2] deduced ...

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...of a hypergraph). It seems of interest to try to obtain similar product structures in the traditional infinitary setting; the argument in [2] achieves this to some extent in the l = 2 case. (See also =-=[25]-=- for another (not entirely satisfactory) attempt to endow dynamical systems with hypergraph structure.) This would likely lead to a more traditional infinitary proof of Theorem 1.1. This paper is orga...

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...L∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra =-=[3]-=- (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal...

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...ich on comparison with Example 1.5 yields (8) in the l = 2 case. Remark 3.20. The above elementary arithmetic manipulations are essentially the same manipluations used in the hypergraph approach (see =-=[21]-=-, [19], [10], [23]) to Szemerédi’s theorem [22] or the Furstenberg-Katznelson theorem [7], in order to rewrite the problem in a “hypergraph” form.12 TERENCE TAO The operator ∆N is clearly linear. For...

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...range [M, F(M)] rather than an infinite range [M, +∞). This allows us to perform pigeonholing tricks based on locating several disjoint intervals of the form [M, F(M)], as was recently carried out in =-=[26]-=-. Indeed our arguments here have some of the “multiscale analysis” flavour of [26]. See also [23], in which functions such as F play a key role in establishing a hypergraph regularity lemma. We shall ...

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...e infinite pigeonhole principle, which is notoriously hard to finitise. Indeed the situation here is somewhat reminiscent of that of the Paris-Harrington theorem [18]. Note that it was established in =-=[28]-=- (see also [20]) that the Lebesgue dominated convergence theorem is equivalent in the reverse mathematics sense to the arithmetic comprehension axiom (ACA), which does strongly suggest that the depend...

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...ng [29] for l = 3 and Frantzikinakis and Kra [3] for general l under the additional hypotheses that each (for i ̸= j) are individually ergodic transformations. The result was also obtained by Lesigne =-=[15]-=- for certian distal systems. In the special case Ti = T i for some measure-preserving transformation T : X → X, this result of the Ti and the TiT −1 j 12 TERENCE TAO was first obtained for general l ...

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...e Gödel functional interpretation of the infinitary statement, which is also closely related to the Kriesel no-counterexample interpretation [13], [14] or Herbrand normal form of such statements; see =-=[12]-=- for further discussion. We thank Ulrich Kohlenbach for pointing out this connection. 2 This is analogous to how the argument in [2] deduced the l = 2 case of Theorem 1.1 from various one-dimensional ...

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...e and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in =-=[29]-=-). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those le...

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...onhole principle, which is notoriously hard to finitise. Indeed the situation here is somewhat reminiscent of that of the Paris-Harrington theorem [18]. Note that it was established in [28] (see also =-=[20]-=-) that the Lebesgue dominated convergence theorem is equivalent in the reverse mathematics sense to the arithmetic comprehension axiom (ACA), which does strongly suggest that the dependence of M ′ ∗,F...

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...rgodic averages 1 PN−1 N n=0 f1(T n 1 x). . . fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1, . . . , fl ∈ L∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne =-=[3]-=- and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [4] (with the l = 3 case of this result established earlier in [30]). Our approa...

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...r 1 In proof theory, this finitisation is known as the Gödel functional interpretation of the infinitary statement, which is also closely related to the Kriesel no-counterexample interpretation [13], =-=[14]-=- or Herbrand normal form of such statements; see [12] for further discussion. We thank Ulrich Kohlenbach for pointing out this connection. 2 This is analogous to how the argument in [2] deduced the l ...