On a problem of countable expansions

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作者：Yuru Zou^a ; ^{yrzou@163.com" class="auth_mail" title="E-mail the corresponding author} ; Derong Kong^b ; ^{derongkong@126.com" class="auth_mail" title="E-mail the corresponding author}
关键词：11A63 ; 37A45
刊名：Journal of Number Theory
出版年：2016
出版时间：January 2016
年：2016
卷：158
期：Complete
页码：134-150
全文大小：346 K

文摘

For a real number q∈(1,2)

q \in (1, 2)

and x∈[0,1/(q−1)]

x \in [0, 1 / (q - 1)]

, the infinite sequence (d_i)

(d_{i})

is called a q-expansion of x if

x = \sum_{i = 1}^{\infty} \frac{d_{i}}{q^{i}}, d_{i} \in {0, 1} for all i \geq 1 .

For m=1,2,鈰?/span>

m = 1, 2, 鈰?/mo>

or ℵ₀

ℵ_{0}

we denote by B_m

B_{m}

the set of q∈(1,2)

q \in (1, 2)

such that there exists x∈[0,1/(q−1)]

x \in [0, 1 / (q - 1)]

having exactly m different q-expansions. It was shown by Sidorov [18] that q₂:=min鈦₂≈1.71064

q_{2} : = \min 鈦?/mo>B_{2}\approx1.71064

, and later asked by Baker [1] whether q₂∈B_ℵ₀

q_{2} \in B_{ℵ_{0}}

? In this paper we provide a negative answer to this question and conclude that B_ℵ₀

B_{ℵ_{0}}

is not a closed set. In particular, we give a complete description of x∈[0,1/(q₂−1)]

x \in [0, 1 / (q_{2} - 1)]

having exactly two different q₂

q_{2}

-expansions.

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