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


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
Full Text: DOI


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