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On the maximal ergodic theorem for certain subsets of the integers. (English) Zbl 0642.28010
Giving an affirmative answer to a problem considered by A. Bellow [Lect. Notes Math. 945, 429-431 (1982)] and H. Furstenberg (Proc. Durham Conf., June 1982), the author proves the following deep and interesting result: Let (X,\(\mu\),T) be a dynamical system then \(\frac{1}{n}\sum_{k\leq n}T^{k^ 2}f\) converges almost surely for any \(f\in L^ 2(X,\mu)\), more generally \(k^ 2\) can be replaced by an arbitrary polynomial function p(k) with integer coefficients. The problem is reduced to the proof of an inequality \(\| M_ nf\|_ 2\leq C\| f\|_ 2,\quad f\in L^ 2(X,\mu),\) where \(M_ nf\) is the “maximal function” \(\sup_{j\leq n}| (\sum_{k\leq j}T^{p(k)})/card\{k: p(k)\leq j\}|.\)
This inequality can be proved by showing the according inequality for the special system (\({\mathbb{Z}},\lambda,T)\), \(\lambda\) the counting measure, T the shift. This can be done by Fourier transform methods and careful estimates of exponential sums (Gauss sums if \(p(k)=k^ 2)\), estimates from A. Sárközy’s paper [Acta Math. Acad. Sci. Hung. 31, 125- 149 (1978; Zbl 0387.10033)] are used, the case \(p(k)=k^ t\) is associated with the Waring problem. The method of major arcs is of fundamental importance, based on I. M. Vinogradov [The method of trigonometric sums in the theory of numbers (1954; Zbl 0055.275; Russian original 1947; Zbl 0041.370)] and R. C. Vaughan [The Hardy- Littlewood method (1981; Zbl 0455.10034)].
As consequences one obtains results on uniform distribution, e.g. \(\frac{1}{n}\sum_{k\leq n}f(x+m^ t\alpha)\to \int f(x)dx\) almost surely for \(\alpha\not\in {\mathbb{Q}}\) and \(f\in L^{\infty}({\mathbb{R}}/{\mathbb{Z}})\) or \(\frac{1}{n}\sum_{m\leq n}f(2^{mt}x)\to \int f(x)dx\) a.s. and \(f\in L^{\infty}({\mathbb{R}}/{\mathbb{Z}})\), generalizing the Riesz-Raikov result for \(t=1.\)
Furthermore the author obtains according results for commuting transformations and pointwise ergodic theorems for random sets for \(f\in L^ p\), \(p>1\).
Reviewer: H.Rindler

MSC:
28D05 Measure-preserving transformations
11L40 Estimates on character sums
42A05 Trigonometric polynomials, inequalities, extremal problems
11K06 General theory of distribution modulo \(1\)
42B25 Maximal functions, Littlewood-Paley theory
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References:
[1] J. Bourgain,Théorèmes ergodiques poncheels pour certains ensembles arithmétiques, C.R. Acad. Sci. Paris305 (1987), 397–402.
[2] J. Bourgain,On the pointwise ergodic theorem on L p for arithmetic sets, Isr. J. Math.61 (1988), 73–84, this issue. · Zbl 0642.28011 · doi:10.1007/BF02776302
[3] A. Bellow,Two Problems, Lecture Notes in Math.945, Springer-Verlag, Berlin, pp. 429–431.
[4] A. Bellow and V. Losert,On sequences of density zero in ergodic theory, Contemp. Math.26 (1984), 49–60. · Zbl 0587.28013
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[10] R. C. Vaughan,The Hardy-Littlewood Method, Cambridge tracts,80 (1981). · Zbl 0455.10034
[11] Vinogradov,The Method of Trigonometrical Sums in the Theory of Numbers, Interscience, New York, 1954. · Zbl 0055.27504
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