arXiv
Open Access
2021
On the Mahler measure of the spectrum of rank one maps
el Houcein el Abdalaoui
Abstrak
We extend partially the Kakutani-Zygmund dichotomy theorem to a class of generalized Riesz-product type measures by proving that the generalized Riesz-product is singular if and only if its Mahler measure is zero. As a consequence, we exhibit a new subclass of rank one maps acting on a finite measure space with singular spectrum. In our proof the $H^p$ theory coming to play. Furthermore, by appealing to a deep result of Bourgain, we prove that the Mahler measure of the spectrum of rank one map with cutting parameter $p_n=O(n^β)$, $β\leq 1$ is zero, and we establish that the integral of the absolute part of any generalized Riesz-product is strictly less than 1. This answer partially a question asked by M. Nadkarni.
Topik & Kata Kunci
Penulis (1)
e
el Houcein el Abdalaoui
Akses Cepat
Informasi Jurnal
- Tahun Terbit
- 2021
- Bahasa
- en
- Sumber Database
- arXiv
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- Open Access ✓