On a problem of countable expansions

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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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=6f4daa17826ae2ff1d7c7a0d62fc59e7" title="Click to view the MathML source">q∈(1,2)class="mathContainer hidden">class="mathCode">

q \in (1, 2)

and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=4a9f79719b7d732ef1c65c1f20675321" title="Click to view the MathML source">x∈[0,1/(q−1)]class="mathContainer hidden">class="mathCode">

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

, the infinite sequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=2dfec23b6c4a5648aa3d31c12caec25a" title="Click to view the MathML source">(d_i)class="mathContainer hidden">class="mathCode">

(d_{i})

is called a q-expansion of x if

class="formula" id="fm0010">

class="mathml">class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si4.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=3d5a5b87a7bace6e82575df5c64b1450">class="imgLazyJSB inlineImage" height="46" width="272" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002309-si4.gif">

class="mathContainer hidden">class="mathCode">

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

class="temp" src="/sd/blank.gif">

For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si5.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=49365d93cbb2d6d97c42e4802b3c006d" title="Click to view the MathML source">m=1,2,鈰?/span>class="mathContainer hidden">class="mathCode">

m = 1, 2, 鈰?/mo>

or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si6.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=6fc3b98c6d28fd9c0dd74af8173f93f1" title="Click to view the MathML source">ℵ₀class="mathContainer hidden">class="mathCode">

ℵ_{0}

we denote by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si7.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=8ccaa2208522d13e6567ea70a3b0e630" title="Click to view the MathML source">B_mclass="mathContainer hidden">class="mathCode">

B_{m}

the set of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=6f4daa17826ae2ff1d7c7a0d62fc59e7" title="Click to view the MathML source">q∈(1,2)class="mathContainer hidden">class="mathCode">

q \in (1, 2)

such that there exists class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=4a9f79719b7d732ef1c65c1f20675321" title="Click to view the MathML source">x∈[0,1/(q−1)]class="mathContainer hidden">class="mathCode">

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

having exactly m different q-expansions. It was shown by Sidorov [18] that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si9.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=f5f82b25a075ab34c0d8235546df2ac9" title="Click to view the MathML source">q₂:=min鈦₂≈1.71064class="mathContainer hidden">class="mathCode">

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

, and later asked by Baker [1] whether class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si10.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=fe0ef6e9565404fecad19864c8ceb190" title="Click to view the MathML source">q₂∈B_ℵ₀class="mathContainer hidden">class="mathCode">

q_{2} \in B_{ℵ_{0}}

? In this paper we provide a negative answer to this question and conclude that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si11.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=3bfb9f22d5a593a64c4fb6680ac2a119" title="Click to view the MathML source">B_ℵ₀class="mathContainer hidden">class="mathCode">

B_{ℵ_{0}}

is not a closed set. In particular, we give a complete description of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si12.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=513d4bbb3d17e2e14dcf2b06e2b00f7b" title="Click to view the MathML source">x∈[0,1/(q₂−1)]class="mathContainer hidden">class="mathCode">

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

having exactly two different class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002309&_mathId=si13.gif&_user=111111111&_pii=S0022314X15002309&_rdoc=1&_issn=0022314X&md5=47cce80b4ac8ea1158da75cec1874969" title="Click to view the MathML source">q₂class="mathContainer hidden">class="mathCode">

q_{2}

-expansions.

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