On the maximal ergodic theorem for certain subsets of the integers
Giving an affirmative answer to a problem considered by \textit{A. Bellow} [Lect. Notes Math. 945, 429-431 (1982)] and \textit{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 \textit{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 \textit{I. M. Vinogradov} [The method of trigonometric sums in the theory of numbers (1954; Zbl 0055.275; Russian original 1947; Zbl 0041.370)] and \textit{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 \(f\in L^ p\), \(p>1\).
scientific article; zbMATH DE number 4046064
scientific article