Baire category and the relative growth rate for partial quotients in continued fractions

Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1...
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Autor*in:	Chang, Xinyi [verfasserIn] Dong, Yihan Liu, Mengchen Shang, Lei

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2023

Schlagwörter:	Baire category Irrationality exponent Residual set Continued fractions

Anmerkung:	© Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Übergeordnetes Werk:	Enthalten in: Archiv der Mathematik - Berlin : Springer, 1948, 122(2023), 1 vom: 26. Sept., Seite 41-46
Übergeordnetes Werk:	volume:122 ; year:2023 ; number:1 ; day:26 ; month:09 ; pages:41-46

Links:	Volltext

DOI / URN:	10.1007/s00013-023-01914-6

Katalog-ID:	SPR054317886

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520			\|a Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$.
650		4	\|a Baire category \|7 (dpeaa)DE-He213
650		4	\|a Irrationality exponent \|7 (dpeaa)DE-He213
650		4	\|a Residual set \|7 (dpeaa)DE-He213
650		4	\|a Continued fractions \|7 (dpeaa)DE-He213
700	1		\|a Dong, Yihan \|4 aut
700	1		\|a Liu, Mengchen \|4 aut
700	1		\|a Shang, Lei \|0 (orcid)0000-0002-7458-3590 \|4 aut
773	0	8	\|i Enthalten in \|t Archiv der Mathematik \|d Berlin : Springer, 1948 \|g 122(2023), 1 vom: 26. Sept., Seite 41-46 \|w (DE-627)253389976 \|w (DE-600)1458441-4 \|x 1420-8938 \|7 nnns
773	1	8	\|g volume:122 \|g year:2023 \|g number:1 \|g day:26 \|g month:09 \|g pages:41-46
856	4	0	\|u https://dx.doi.org/10.1007/s00013-023-01914-6 \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4277
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4393
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allfields	10.1007/s00013-023-01914-6 doi (DE-627)SPR054317886 (SPR)s00013-023-01914-6-e DE-627 ger DE-627 rakwb eng Chang, Xinyi verfasserin aut Baire category and the relative growth rate for partial quotients in continued fractions 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213 Dong, Yihan aut Liu, Mengchen aut Shang, Lei (orcid)0000-0002-7458-3590 aut Enthalten in Archiv der Mathematik Berlin : Springer, 1948 122(2023), 1 vom: 26. Sept., Seite 41-46 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:122 year:2023 number:1 day:26 month:09 pages:41-46 https://dx.doi.org/10.1007/s00013-023-01914-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 122 2023 1 26 09 41-46
spelling	10.1007/s00013-023-01914-6 doi (DE-627)SPR054317886 (SPR)s00013-023-01914-6-e DE-627 ger DE-627 rakwb eng Chang, Xinyi verfasserin aut Baire category and the relative growth rate for partial quotients in continued fractions 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213 Dong, Yihan aut Liu, Mengchen aut Shang, Lei (orcid)0000-0002-7458-3590 aut Enthalten in Archiv der Mathematik Berlin : Springer, 1948 122(2023), 1 vom: 26. Sept., Seite 41-46 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:122 year:2023 number:1 day:26 month:09 pages:41-46 https://dx.doi.org/10.1007/s00013-023-01914-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 122 2023 1 26 09 41-46
allfields_unstemmed	10.1007/s00013-023-01914-6 doi (DE-627)SPR054317886 (SPR)s00013-023-01914-6-e DE-627 ger DE-627 rakwb eng Chang, Xinyi verfasserin aut Baire category and the relative growth rate for partial quotients in continued fractions 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213 Dong, Yihan aut Liu, Mengchen aut Shang, Lei (orcid)0000-0002-7458-3590 aut Enthalten in Archiv der Mathematik Berlin : Springer, 1948 122(2023), 1 vom: 26. Sept., Seite 41-46 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:122 year:2023 number:1 day:26 month:09 pages:41-46 https://dx.doi.org/10.1007/s00013-023-01914-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 122 2023 1 26 09 41-46
allfieldsGer	10.1007/s00013-023-01914-6 doi (DE-627)SPR054317886 (SPR)s00013-023-01914-6-e DE-627 ger DE-627 rakwb eng Chang, Xinyi verfasserin aut Baire category and the relative growth rate for partial quotients in continued fractions 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213 Dong, Yihan aut Liu, Mengchen aut Shang, Lei (orcid)0000-0002-7458-3590 aut Enthalten in Archiv der Mathematik Berlin : Springer, 1948 122(2023), 1 vom: 26. Sept., Seite 41-46 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:122 year:2023 number:1 day:26 month:09 pages:41-46 https://dx.doi.org/10.1007/s00013-023-01914-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 122 2023 1 26 09 41-46
allfieldsSound	10.1007/s00013-023-01914-6 doi (DE-627)SPR054317886 (SPR)s00013-023-01914-6-e DE-627 ger DE-627 rakwb eng Chang, Xinyi verfasserin aut Baire category and the relative growth rate for partial quotients in continued fractions 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213 Dong, Yihan aut Liu, Mengchen aut Shang, Lei (orcid)0000-0002-7458-3590 aut Enthalten in Archiv der Mathematik Berlin : Springer, 1948 122(2023), 1 vom: 26. Sept., Seite 41-46 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:122 year:2023 number:1 day:26 month:09 pages:41-46 https://dx.doi.org/10.1007/s00013-023-01914-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 122 2023 1 26 09 41-46
language	English
source	Enthalten in Archiv der Mathematik 122(2023), 1 vom: 26. Sept., Seite 41-46 volume:122 year:2023 number:1 day:26 month:09 pages:41-46
sourceStr	Enthalten in Archiv der Mathematik 122(2023), 1 vom: 26. Sept., Seite 41-46 volume:122 year:2023 number:1 day:26 month:09 pages:41-46
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Baire category Irrationality exponent Residual set Continued fractions
isfreeaccess_bool	false
container_title	Archiv der Mathematik
authorswithroles_txt_mv	Chang, Xinyi @@aut@@ Dong, Yihan @@aut@@ Liu, Mengchen @@aut@@ Shang, Lei @@aut@@
publishDateDaySort_date	2023-09-26T00:00:00Z
hierarchy_top_id	253389976
id	SPR054317886
language_de	englisch
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. 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author	Chang, Xinyi
spellingShingle	Chang, Xinyi misc Baire category misc Irrationality exponent misc Residual set misc Continued fractions Baire category and the relative growth rate for partial quotients in continued fractions
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topic_title	Baire category and the relative growth rate for partial quotients in continued fractions Baire category (dpeaa)DE-He213 Irrationality exponent (dpeaa)DE-He213 Residual set (dpeaa)DE-He213 Continued fractions (dpeaa)DE-He213
topic	misc Baire category misc Irrationality exponent misc Residual set misc Continued fractions
topic_unstemmed	misc Baire category misc Irrationality exponent misc Residual set misc Continued fractions
topic_browse	misc Baire category misc Irrationality exponent misc Residual set misc Continued fractions
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	Baire category and the relative growth rate for partial quotients in continued fractions
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title_full	Baire category and the relative growth rate for partial quotients in continued fractions
author_sort	Chang, Xinyi
journal	Archiv der Mathematik
journalStr	Archiv der Mathematik
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container_start_page	41
author_browse	Chang, Xinyi Dong, Yihan Liu, Mengchen Shang, Lei
container_volume	122
format_se	Elektronische Aufsätze
author-letter	Chang, Xinyi
doi_str_mv	10.1007/s00013-023-01914-6
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title_sort	baire category and the relative growth rate for partial quotients in continued fractions
title_auth	Baire category and the relative growth rate for partial quotients in continued fractions
abstract	Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstractGer	Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstract_unstemmed	Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$. © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
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title_short	Baire category and the relative growth rate for partial quotients in continued fractions
url	https://dx.doi.org/10.1007/s00013-023-01914-6
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author2	Dong, Yihan Liu, Mengchen Shang, Lei
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doi_str	10.1007/s00013-023-01914-6
up_date	2024-07-04T01:02:14.291Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">SPR054317886</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20240110064631.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">240110s2023 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00013-023-01914-6</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR054317886</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00013-023-01914-6-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Chang, Xinyi</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Baire category and the relative growth rate for partial quotients in continued fractions</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2023</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let %$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]%$ be the continued fraction expansion of an irrational number %$x\in (0,1)%$, and %$q_n(x)%$ be the denominator of its n-th convergent. In this note, the Baire category of the set E(α,β):=x∈(0,1)\Q:lim infn→∞logan+1(x)logqn(x)=α,lim supn→∞logan+1(x)logqn(x)=β%$\begin{aligned} E(\alpha ,\beta ):= & {} \left\{ x\in (0,1)\backslash \mathbb {Q}:\liminf _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\alpha ,\right. \\ {}{} & {} \left. \quad \limsup _{n \rightarrow \infty }\frac{\log a_{n+1}(x)}{\log q_n(x)} =\beta \right\} \end{aligned}%$for %$\alpha ,\beta \in [0,\infty ]%$ with %$\alpha \le \beta %$ is studied. We prove that the set %$E(\alpha ,\beta )%$ is residual if and only if %$\alpha =0%$ and %$\beta =\infty %$.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Baire category</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Irrationality exponent</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Residual set</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Continued fractions</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Dong, Yihan</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Liu, Mengchen</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Shang, Lei</subfield><subfield code="0">(orcid)0000-0002-7458-3590</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Archiv der Mathematik</subfield><subfield code="d">Berlin : Springer, 1948</subfield><subfield code="g">122(2023), 1 vom: 26. 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