A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages

Abstract

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.

Keywords:

• Van der Corput
• recurrence
• noncommutative ergodic theory

2020 Mathematics Subject Classifications:

• Primary: 47A35
• 37A30
• Secondary:37B20

Acknowledgements

I would like to thank Srivatsa Srinivas for helpful discussions regarding Fourier analysis that lead to significant improvements in this paper. I would also like to thank the referees for their careful reading of the paper as well as their useful comments which lead to further improvements.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We work with $f\in L_0^2(X,\mu )$ instead of $L^2(X,\mu )$ so that ${\mu }_R$ has a pointmass at $\left\{0\right\}$ if and only if R is not ergodic.

2 A finitely generated solvable group has exponential growth if and only if it contains no nilpotent subgroups of finite index, so the assumption of exponential growth ensures that the group G is not virtually nilpotent. We also recall that Berend's example involved a non-solvable group action.

3 See [16, Theorem 2.4.30] for a statement of the uniform distribution properties possessed by such sequences.

4 While we have not shown that the limits in Equation (12(12) $\underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{n=1}^N{T}^{n}f\cdot S^{p\left(n\right)}g$(12) ) exist when $k_n=p\left(n\right)$, this is easy to deduce after replacing g with $\mathbb{E}\left[g|{\mathcal{K}}_{\mathrm{r}\mathrm{a}\mathrm{t}}\right(S\left)\right]$. See also Remark 1.17.

5 However, in the case of Theorem 1.8 we would have to strengthen the assumption of uniform distribution of $\left(\right(k_{n+h}-k_n)\alpha {)}_{n=1}^{\mathrm{\infty }}$ to the assumption of well distribution.

Additional information

Funding

I also acknowledge being supported by grant 2019/34/E/ST1/00082 for the project ‘Set theoretic methods in dynamics and number theory’, NCN (The National Science Centre of Poland).