## 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
Full Text:

### 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 [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 [5] H. Furstenberg, Proc. Durham Conf., June 1982. [6] R. Lidl, H. and Neiderreiter,Finite fields, Encyclopedia of Mathematics and its Applications, 20, Addison-Wesley Publ. Co., 1983. [7] J. M. Marstrand,On Khinchine’s conjecture about strong uniform distribution, Proc. London Math. Soc.21 (1970), 540–556. · Zbl 0208.31402 [8] A. Sarközy,On difference sets of sequences of integers, I, Acta Math. Acad. Sci. Hung.31 (1978), 125–149. · Zbl 0387.10033 [9] E. Stein,Beijing Lectures in Harmonic Analysis, Ann. Math. Studies, Princeton University Press, 1986, p. 112. · Zbl 0595.00015 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.