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The Arithmetic Version of the Frequency Transition Conjecture: New Proof and Generalization

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Abstract

The arithmetic version of the frequency transition conjecture for the almost Mathieu operators was recently proved by Jitomirskaya and Liu [34]. We give a new proof via reducibility theory and duality, which derives from the method developed in [22] (in fact it is a simplified version). This new proof is applicable to more general quasiperiodic operators.

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Notes

  1. Pure point spectrum with exponentially decaying eigenfunctions.

  2. Denote by

    $$\begin{aligned} \delta (\alpha ,\theta )=\limsup \limits _{k\rightarrow \infty }-\frac{\ln \Vert 2\theta +k\alpha \Vert _{{{\mathbb {R}}}/{{\mathbb {Z}}}}}{|k|}. \end{aligned}$$

    The original conjecture is for the set of \(\theta\) with \(\delta (\alpha ,\theta )=0\) which contains a zero Lebesgue measure set more than the set of \(\alpha\)-Diophantine \(\theta\).

  3. We refer readers to Lemma 3.2 in [22] for the proof of the second inequality.

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Acknowledgements

J. You was partially supported by National Key R&D Program of China (No. 2020YFA0713300) and NSFC (No. 11871286). L. Ge and X. Zhao were partially supported by NSF DMS-1901462. L. Ge was partially supported by AMS-Simons Travel Grant 2020-2022. X. Zhao was partially supported by China Scholarship Council (No. 201906190072) and NSFC (No. 11771205).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Lingrui Ge & Xin Zhao

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Jiangong You

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  1. Lingrui Ge
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Correspondence to Jiangong You.

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Ge, L., You, J. & Zhao, X. The Arithmetic Version of the Frequency Transition Conjecture: New Proof and Generalization. Peking Math J 5, 349–364 (2022). https://doi.org/10.1007/s42543-021-00040-y

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