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Note to “Singular continuous Floquet operator for systems with increasing gaps”
J. Math. Anal. Appl. 289 (2004) 722–723 www.elsevier.com/locate/jmaa
Corrigendum
Note to “Singular continuous Floquet operator for systems with increasing gaps” [J. Math. Anal. Appl. 276 (2002) 28–39] ✩ Olivier Bourget Pontificia Universidad Católica, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, CP 690 4411, Macul, Santiago, Chile
This corrigendum aims at filling a gap left in the proof of Lemma 3.2 on pages 34– 35. The major part of this proof remains unchanged except the last sentence. Following the same notations, the reader should replace “It follows from Theorem 12 in [1] that necessarily QN qr,N 2(N−1)/2 for sufficiently large N , which contradicts the first inequality of (3.9),” with the following:
“The sequence (qr,N )NN0 is necessarily increasing. Since ar is irrational, lim qr,N = +∞.
N→+∞
It is therefore possible to extract a subsequence (Nk )k∈N in such a way that ∀k ∈ N, Nk+1 = inf{N > Nk , qNk < qN }. This means that the subsequence (qNk )k∈N is strictly increasing and that each term of the subsequence (pr,Nk /qr,Nk )k∈N is a convergent different from the previous ones. It follows from inequality (3.9) and Theorem 13 [1] that ∀k ∈ N, ∀N ∈ {Nk , . . . , Nk+1 − 1}, 1 pN < ar − N −r+ν . qN 2qN2 k+1 In particular, ∀k ∈ N, 1 2qN2 k+1 ✩
< (Nk+1 − 1)ν−r .
PII of original article: S0022-247X(02)00277-9. E-mail address: [email protected].
0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.08.020
(0.1)
O. Bourget / J. Math. Anal. Appl. 289 (2004) 722–723
723
ν ; i.e., q 2 2ν By hypothesis, ∀k ∈ N, qNk+1 Nk+1 Nk+1 Nk+1 . Therefore, multiplying this inequality with inequality (0.1) entails ∀k ∈ N, 1 −r+3ν −1 −r+ν Nk+1 1 − Nk+1 . 2 The contradiction follows by taking the limit
lim Nk = +∞
k→+∞
and from the fact that r > 3ν > ν.” I am indebted to Andrej Zlatos for pointing out this gap in the proof of Lemma 3.2 and for some informative discussions about it.
Reference [1] A.Y. Khinchin, Continued Fractions, Phoenix Books, 1964.
Corrigendum
Note to “Singular continuous Floquet operator for systems with increasing gaps” [J. Math. Anal. Appl. 276 (2002) 28–39] ✩ Olivier Bourget Pontificia Universidad Católica, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, CP 690 4411, Macul, Santiago, Chile
This corrigendum aims at filling a gap left in the proof of Lemma 3.2 on pages 34– 35. The major part of this proof remains unchanged except the last sentence. Following the same notations, the reader should replace “It follows from Theorem 12 in [1] that necessarily QN qr,N 2(N−1)/2 for sufficiently large N , which contradicts the first inequality of (3.9),” with the following:
“The sequence (qr,N )NN0 is necessarily increasing. Since ar is irrational, lim qr,N = +∞.
N→+∞
It is therefore possible to extract a subsequence (Nk )k∈N in such a way that ∀k ∈ N, Nk+1 = inf{N > Nk , qNk < qN }. This means that the subsequence (qNk )k∈N is strictly increasing and that each term of the subsequence (pr,Nk /qr,Nk )k∈N is a convergent different from the previous ones. It follows from inequality (3.9) and Theorem 13 [1] that ∀k ∈ N, ∀N ∈ {Nk , . . . , Nk+1 − 1}, 1 pN < ar − N −r+ν . qN 2qN2 k+1 In particular, ∀k ∈ N, 1 2qN2 k+1 ✩
< (Nk+1 − 1)ν−r .
PII of original article: S0022-247X(02)00277-9. E-mail address: [email protected].
0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.08.020
(0.1)
O. Bourget / J. Math. Anal. Appl. 289 (2004) 722–723
723
ν ; i.e., q 2 2ν By hypothesis, ∀k ∈ N, qNk+1 Nk+1 Nk+1 Nk+1 . Therefore, multiplying this inequality with inequality (0.1) entails ∀k ∈ N, 1 −r+3ν −1 −r+ν Nk+1 1 − Nk+1 . 2 The contradiction follows by taking the limit
lim Nk = +∞
k→+∞
and from the fact that r > 3ν > ν.” I am indebted to Andrej Zlatos for pointing out this gap in the proof of Lemma 3.2 and for some informative discussions about it.
Reference [1] A.Y. Khinchin, Continued Fractions, Phoenix Books, 1964.