Nilsequences and Multiple Correlations along Subsequences

Le, Anh Ngoc

doi:https://doi.org/10.21985/n2-kv9a-vr02

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Nilsequences and Multiple Correlations along Subsequences

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Arising from the study of multiple ergodic averages, nilsequences and multiple correlation sequences lie at the crossroads of ergodic theory, combinatorics and number theory. We study these types of sequences along various subsequences of integers, and provide applications to ergodic theory and harmonic analysis. Our first result involves multiple correlation sequences related to powers of a single transformation. Among other things, we show that for a multiple correlation sequence of this type, say $(\alpha(n))_{n \in \mathbb{N}}$, there exists a uniform limit of nilsequences $(\psi(n))_{n \in \mathbb{N}}$ such that \[ \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N |\alpha(r_n) -\psi(r_n)| = 0 \] for $r_n = Q(n), Q(p_n)$ or $\lfloor n^c \rfloor$ where $Q \in \Z[x]$ non-constant, $p_n$ is n-th prime and $c > 0$. The second result concerns general multiple correlation sequences of several commuting transformations. We show that for every sequence $(\beta(n))_{n \in \mathbb{N}}$ of this type and every $\epsilon > 0$, there exists a nilsequence $(\psi(n))_{n \in \mathbb{N}}$ such that \[ \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N |\beta(p_n) - \psi(p_n)| \leq \epsilon. \] Next we study interpolation sets for nilsequences. In particular, we investigate almost periodic sequences ($1$-step nilsequences) when evaluated along sparse sets. Strzelecki proved that lacunary sets are interpolation sets for almost periodic sequences. Our result shows that the same is false for any sublacunary set. Finally, as applications of the first decomposition result, we prove optimal lower bounds for multiple recurrence along the sequences of primes, Hardy field sequences and Beatty sequences.

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