On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II
We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or , using the entropy method. It follows that such a chain with positive lower density is bad for . There also exist such bad chains with zero density.
We show that for any irrational number α and a sequence of integers such that , there exists a continuous measure μ on the circle such that . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence of integers such that and such that is dense on the circle if and only if θ ∉ ℚα + ℚ.
If denotes the sequence of best approximation denominators to a real , and denotes the sum of digits of in the digit representation of to base , then for all irrational, the sequence is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if has bounded continued fraction coefficients.
In this paper, we give a new upper-bound for the discrepancyfor the sequence , when and .