Let \((\Omega, {\mathcal B},\mu)\) be a probability space, \(T\) a measure- preserving automorphism, \(p(x)\) a polynomial with integer coefficients. Then, generalizing earlier results [the author, Isr. J. Math. 61, No. 1, 39–72 (1988; Zbl 0642.28010); ibid. 73–84 (1988; Zbl 0642.28011)], the author proves the following beautiful and deep pointwise ergodic theorem: \((1/N)\sum_{1\leq n\leq N}T^{p(n)}f\) converges almost surely for all \(f\in L^ p(\Omega,\mu),\) for every \(p>1.\) The case \(p=1\) remains open, but the methods give also a new and simple proof of Birkhoff’s ergodic theorem [cf. the author, Lect. Notes Math. 1317, 204–223 (1988; Zbl 0662.47006)]. The methods given here also allow a generalization of the result to other sequences, e.g. \([p(n)]\), for real polynomials \(p(n)\) (proved only for bounded measurable functions), or the sequence of prime numbers, earlier shown by M. Wierdl [Isr. J. Math. 64, No. 3, 315–336 (1988; Zbl 0695.28007)]. For \(p=2\) this was settled also by the author in the third paper cited above. The results also generalize to non-invertible isometries on \(L^ p\), \(p>1\) (e.g. \(2^{p(n)}\); the case \(p(x)=x\) implies the classical result of Riesz and Raikov, which cannot be generalized to \(L^ 1\) by Marstrand’s counterexample to the Khintchine conjecture).
The author has an appendix answering an open problem on return times, in joint work with Furstenberg, Katznelson and Ornstein: Let \((X, {\mathcal B},\mu,T)\) be an ergodic system, \(A\) a set of positive measure, \(x\in X,\quad \Lambda_ x=\{n\in {\mathbb Z}_+,\quad T^ nx\in A\}.\) Then \((1/N)\sum_{1\leq n\leq N,n\in \Lambda_ x}S^ ng\) converges almost surely for any measure-preserving system \((Y,D,\nu,S)\), and \(g\in L^ 1(Y)\), i.e. almost all sequences \(\Lambda_ x\) satisfy the pointwise ergodic theorem (improving the mean ergodic version of Wiener and Wintner). The methods of proofs are very interesting (cf. the reviews of the papers cited above). In this paper, in addition an inequality on martingales of D. Lépingle [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36, 295–316 (1976; Zbl 0325.60047)] is of essential importance.