## Almost sure convergence and bounded entropy.(English)Zbl 0677.60042

Let ($${\mathcal X},\mu)$$ be a probability space, $$S_ n$$ a sequence of operators on $$L^ 2(\mu)$$, $$\| S_ n\| \leq 1$$, $$T_ j$$ a sequence of positive isometric operators satisfying $$T_ j(1)=1$$, $$J^{-1}\sum_{j\leq J}T_ jf\to \int f d\mu$$ $$\forall f\in L^ 1$$ and $$T_ jS_ n=S_ nT_ j$$. Using the theory of Gaussian processes it is proved that if $$S_ nf$$ converges almost surely for all $$f\in L^ p$$, $$p<\infty$$ $$(p=\infty)$$, then there is a uniform entropy estimate: $$\delta (\log N_ f(\delta))^{1/2}<C,$$ $$\delta >0$$ $$(N_ f(\delta)\leq C(\delta))$$ for all $$f\in L^ 2$$, $$\| f\|_ 2\leq 1$$. $$N_ f(\delta)$$ denotes the number of $$\delta$$-balls needed for covering $$\{S_ nf$$, $$n\in {\mathbb{N}}\}.$$
As a consequence one obtains new proofs for Rudin’s solution of the Marcinkiewicz-Zygmund problem for Riemann sums [W. Rudin, Proc. Am. Math. Soc. 15, 321-324 (1964; Zbl 0132.036)], or Marstrand’s negative solution of Khintchine’s conjecture. Even a weaker version of Khintchine’s problem (considered by Erdős) is disproved. It follows even more generally that for any sequence of decreasing numbers $$\lambda_ j>0$$, $$\sum \lambda_ j=\infty$$, there is a function $$f\in L^{\infty}({\mathbb{R}}/{\mathbb{Z}})$$ such that $$(1/\sigma_ n)\sum_{j\leq n}\lambda_ jf(jx)$$ does not converge almost surely $$(\sigma_ n=\sum_{j\leq n}\lambda_ j).$$
As a further application one obtains an affirmative answer to a question of A. Bellow, Two problems. Measure theory, Proc. Conf., Oberwolfach 1981, Lect. Notes Math. 945, 429 (1987): Let $$(a_ j)$$ be any sequence converging to 0 $$(a_ n\neq 0$$ $$\forall_ j)$$. Then there exists $$f\in L^ 1({\mathbb{R}}/{\mathbb{Z}})$$ (f can be even choosen to be the characteristic function of a measurable set) such that $$n^{- 1}\sum_{j\leq n}f(x+a_ j)$$ does not converge to f almost everywhere.
Reviewer: H.Rindler

### MSC:

 60G15 Gaussian processes 22D40 Ergodic theory on groups 28D20 Entropy and other invariants 11K06 General theory of distribution modulo $$1$$ 40H05 Functional analytic methods in summability

Zbl 0132.036
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### References:

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