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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:
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