On the maximal ergodic theorem for certain subsets of the integers. (English) Zbl 0642.28010

Bourgain, J.

Isr. J. Math. 61, No. 1, 39-72 (1988).

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

Keywords:

maximal ergodic theorem; maximal function; exponential sums; Gauss sums; Waring problem; uniform distribution; pointwise ergodic theorems

Citations:

Zbl 0642.28011; Zbl 0387.10033; Zbl 0055.275; Zbl 0041.370; Zbl 0455.10034

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