# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  A. Bellow,Two problems, Proc. Oberwolfach Conference on Measure Theory (June 1987), Springer Lecture Notes in Math.945, 1987.  R. M. Dudley,Sample functions of the Gaussian process, Ann. Probab.1 (1973), 66–103. · Zbl 0261.60033  P. Erdös, Lecture, Louisiana State University, November 1987.  B. Jessen,On the approximation of Lebesgue integrals by Riemann sums, Ann. of Math. (2)35 (1934), 248–251. · JFM 60.0209.02  A. Khintchine,Ein Satz über Kettenbruche mit arithmetischen Anwendungen, Math. Z.18 (1923), 289–306. · JFM 49.0159.03  J. F. Koksma,A diophantine property of some summable functions, J. Indian Math. Soc. (N.S.)15 (1951), 87–96. · Zbl 0046.04802  J. Marcinkiewicz and A. Zygmund,Mean values of trigonometrical polynomials, Fund. Math.28 (1937), 131–166. · JFM 63.0221.03  J. M. Marstrand,On Khintchine’s conjecture about strong uniform distribution, Proc. London Math. Soc.21 (1970), 540–556. · Zbl 0208.31402  W. Rudin,An arithmetic property of Riemann sums, Proc. Am. Math. Soc.15 (1964), 321–324. · Zbl 0132.03601  E. M. Stein,On limits of sequences of operators, Ann. Math.74 (1961), 140–170. · Zbl 0103.08903
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.