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Sublacunary sets and interpolation sets for nilsequences

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  • A set $ E \subset \mathbb{N} $ is an interpolation set for nilsequences if every bounded function on $ E $ can be extended to a nilsequence on $ \mathbb{N} $. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here $ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $ with $ r_1 < r_2 < \ldots $ is sublacunary if $ \lim_{n \to \infty} (\log r_n)/n = 0 $. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for $ 2 $-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.

    Mathematics Subject Classification: Primary: 37A46; Secondary: 37A44.


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