Almost sure convergence and bounded entropy
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 [\textit{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 \textit{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.
sequence of positive isometric operators
Marcinkiewicz-Zygmund problem for Riemann sums
scientific article; zbMATH DE number 4109749
scientific article