Abstract
We show that some sequences of real numbers involving sharp normal numbers or non-Liouville numbers are uniformly distributed modulo 1. In particular, we prove that if \(\tau (n)\) stands for the number of divisors of n and \(\alpha \) is a binary sharp normal number, then the sequence \((\alpha \tau (n))_{n\ge 1}\) is uniformly distributed modulo 1 and that if g(x) is a polynomial of positive degree with real coefficients and whose leading coefficient is a non-Liouville number, then the sequence \((g(\tau (\tau (n))))_{n \ge 1}\) is also uniformly distributed modulo 1.
Résumé
Nous montrons que certaines suites de nombres réels impliquant des nombres normaux robustes et des nombres non-Liouville sont uniformément réparties modulo 1. En particulier, nous démontrons que si \(\tau (n)\) représente le nombre de diviseurs de n, alors, étant donné un nombre normal binaire robuste \(\alpha \), la suite correspondante \((\alpha \tau (n))_{n\ge 1}\) est uniformément répartie modulo 1 et nous démontrons également que si g(x) est un polynôme à coefficients réels de degré positif et dont le coefficient principal est un nombre non-Liouville, alors la suite \((g(\tau (\tau (n))))_{n \ge 1}\) est uniformément répartie modulo 1.
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Acknowledgements
The authors would like to thank the referee for pointing out corrections and also for providing valuable suggestions. The research of the first author was supported in part by a grant from NSERC.
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De Koninck, JM., Kátai, I. On properties of sharp normal numbers and of non-Liouville numbers. Ann. Math. Québec 42, 31–47 (2018). https://doi.org/10.1007/s40316-017-0080-3
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DOI: https://doi.org/10.1007/s40316-017-0080-3