On the number of nonzero digits in the beta-expansions of algebraic numbers

Kaneko, Hajme

doi:10.4171/RSMUP/136-14

Kaneko, Hajme ¹

¹ University of Tsukuba, TSUKUBA, IBARAKI, JAPAN

Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 205-223.

Résumé

Many mathematicians have investigated the base- $b$ expansions for integral base- $b \geq 2$ , and more general $β$ -expansions for a real number $β > 1$ . However, little is known on the $β$ -expansions of algebraic numbers. The main purpose of this paper is to give new lower bounds for the numbers of nonzero digits in the $β$ -expansions of algebraic numbers under the assumption that $β$ is a Pisot or Salem number. As a consequence of our main results, we study the arithmetical properties of power series $\sum_{n = 1}^{\infty} β^{- κ (z; n)}$ , where $z > 1$ is a real number and $κ (z; n) = ⌊ n^{z} ⌋$ .

DOI : 10.4171/RSMUP/136-14

Classification : 11, 05, 14
Mots clés : Beta expansions, nonzero digits, Pisot numbers, Salem numbers

Affiliations des auteurs :

Kaneko, Hajme ¹

¹ University of Tsukuba, TSUKUBA, IBARAKI, JAPAN

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     title = {On the number of nonzero digits in the beta-expansions of algebraic numbers},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {205--223},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {136},
     year = {2016},
     doi = {10.4171/RSMUP/136-14},
     url = {http://archive.numdam.org/articles/10.4171/RSMUP/136-14/}
}

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Kaneko, Hajme. On the number of nonzero digits in the beta-expansions of algebraic numbers. Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 205-223. doi : 10.4171/RSMUP/136-14. http://archive.numdam.org/articles/10.4171/RSMUP/136-14/

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