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On odd-normal numbers

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Abstract

A real number x is considered normal in an integer base \(b \geqslant 2\) if its digit expansion in this base is “equitable”, ensuring that for each \(k \geqslant 1\), every ordered sequence of k digits from \(\{0, 1, \ldots , b-1\}\) occurs in the digit expansion of x with the same limiting frequency. Borel’s classical result [4] asserts that Lebesgue-almost every \(x \in {\mathbb {R}}\) is normal in every base \(b \geqslant 2\). This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension [30] but zero Fourier dimension. The latter condition means that \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\) supports a Rajchman measure \(\mu \), whose Fourier transform \({\widehat{\mu }}(\xi )\) approaches 0 as \(|\xi | \rightarrow \infty \) by definiton, albeit slower than any negative power of \(|\xi |\). Moreover, the decay rate of \({\widehat{\mu }}\) is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt [38] and a construction of Lyons [24]. As a consequence, \(\mathscr {N}({\mathscr {O}}, {\mathscr {E}})\) emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem [17] in the special case of \({\mathscr {N}}({\mathscr {O}}, {\mathscr {E}})\).

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Acknowledgements

The authors thank Dr. Xiang Gao for introducing them to the area of metrical number theory, for valuable discussions and extensive references on the subject. This work was initiated in 2022, when JZ was visiting University of British Columbia on a study leave from China University of Mining and Technology-Beijing, funded by a scholarship from China Scholarship Council. He would like to thank all three organizations for their support that enabled his visit. JZ was also supported by National Natural Science Foundation of China (Grant nos. 11801555, 11971058 and 12071431). MP was partially supported by a Discovery grant from Natural Sciences and Engineering Research Council of Canada (NSERC).

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, V6T 1Z2, Canada

    Malabika Pramanik

  2. School of Science, China University of Mining and Technology-Beijing, Beijing, 100083, P. R. China

    Junqiang Zhang

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  1. Malabika Pramanik
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Correspondence to Malabika Pramanik.

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Communicated by Apoorva Khare, Rahul Roy.

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Pramanik, M., Zhang, J. On odd-normal numbers. Indian J Pure Appl Math 55, 974–998 (2024). https://doi.org/10.1007/s13226-024-00642-z

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