Some results on non-linear recurrence

Abstract

We study the behavior of measure-preserving systems with continuous time along sequences of the form {n ^α}n∈#x2115;} where α is a positive real number¹. Let {S ^t} _t∈ℝ be an ergodic continuous measure preserving flow on a probability Lebesgue space (X, β, μ). Among other results we show that:

• (1) For all but countably many α (in particular, for all α∈ℚ∖ℤ) one can find anL ^∞-functionf for which the averagesA _N (f)(1/N)=Σ ^N _n=1 f(S ^nα x) fail to converge almost everywhere (the convergence in norm holds for any α!).

• (2) For any non-integer and pairwise distinct numbers α₁, α₂,..., α _k ∈(0, 1) and anyL ^∞-functionsf ₁,f ₂, ...,f _k , one has

  $$\mathop {lim}\limits_{N \to \infty } \left\| {\frac{1}{N}\sum\limits_{n - 1}^N {\prod\limits_{i - 1}^k {f_i (S^{n^{\alpha _i } } x) - \prod\limits_{i - 1}^k {\int_X {f_i d\mu } } } } } \right\|_{L^2 } = 0$$

We also show that Furstenberg’s correspondence principle fails for ℝ-actions by demonstrating that for all but a countably many α>0 there exists a setE⊂ℝ having densityd(E)=1/2 such that, for alln∈ℕ,

$$d(E \cap (E - n^\alpha )) = 0$$

.