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arXiv · 2307.07213

On the maximal spectral type of nilsystems

Ethan Ackelsberg·Florian K. Richter·Or Shalom
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Abstract

Let $(G/\Gamma,R_a)$ be an ergodic $k$-step nilsystem for $k\geq 2$. We adapt an argument of Parry to show that $L^2(G/\Gamma)$ decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host, Kra and Maass for $2$-step nilsystems and a result by Stepin for nilsystems $G/\Gamma$ with connected, simply connected $G$.

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Ethan Ackelsberg, Florian K. Richter, Or Shalom. 2023-07-14. On the maximal spectral type of nilsystems. https://arxiv.org/abs/2307.07213

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