\documentclass[JEP,XML,SOM,Unicode]{cedram}
\datereceived{2022-05-31}
\dateaccepted{2022-09-26}
\dateepreuves{2022-10-03}
\TralicsDefs{\addattributestodocument{type}{corrigendum}}
\Relation{10.5802/jep.146}

\theoremstyle{plain}
\newtheorem*{prop*}{Proposition}

\let\epsilon\varepsilon
\let\Om\Omega

\newcommand\bom{{\boldsymbol a}}
\newcommand\bq{{\boldsymbol q}}
\newcommand\Fre{\mathfrak{E}}
\renewcommand{\P}{\mathbb{P}}

\begin{document}
\frontmatter
\title[Corrigendum to ``Hölder regularity for the spectrum of translation flows'']{Corrigendum to\\ ``Hölder regularity for the spectrum of translation flows''}

\author[\initial{A.} \middlename {I.} \lastname{Bufetov}]{\firstname{Alexander} \middlename {I.} \lastname{Bufetov}}
\address{Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373,\\
39 rue F. Joliot Curie, Marseille, France\\
Steklov Mathematical Institute of RAS, Moscow, Russia\\
Institute for Information Transmission Problems, Moscow, Russia}
\email{bufetov@mi.ras.ru}

\author[\initial{B.} \lastname{Solomyak}]{\firstname{Boris} \lastname{Solomyak}}
\address{Department of Mathematics, Bar-Ilan University\\
Ramat-Gan, Israel}
\email{bsolom3@gmail.com}
\urladdr{https://u.math.biu.ac.il/~solomyb/personal.html}

\begin{abstract}
The paper corrects the formulation of \cite[Prop.\,5.4]{BuSo21}, whose proof is unchanged.
\end{abstract}

\subjclass{37A30, 37B1, 37E35, 28A78}

\keywords{Translation flows, spectral measures, matrix Riesz products, upper Lyapunov exponents, Erd\H{o}s-Kahane argument, Bratteli-Vershik automorphisms, renormalization cocycle}

\altkeywords{Flots de translation, mesures spectrales, produits de Riesz matriciels, exposants de Liapounoff supérieurs, l'argument d'Erd\H{o}s-Kahane, automorphismes de Bratteli-Vershik, cocycle de renormalisation}

\alttitle{Correction à \og Régularité Hölder pour le spectre des flots de translation\fg}

\begin{altabstract}
Cet article corrige la formulation de \cite[Prop.\,5.4]{BuSo21}, dont la preuve est inchangée.
\end{altabstract}

\maketitle
\mainmatter
In \cite[Prop.\,5.4]{BuSo21} a different order of quantifiers was needed from the one stated. The following proposition was, in fact, proven in the paper, and this is the correct formulation:

\begin{prop*}
For any $\epsilon_1>0$ there exist $\delta_0>0$ and $\varrho>0$ such that for $\P$\nobreakdash-a.e.\ $\bom\in\nobreak \Om_\bq$ and any $\delta\in (0,\delta_0)$, for all $B>1$, and every
Oseledets subspace $H_\bom^J$ corresponding to $\bom$, containing the unstable subspace $E^u_\bom$,
\[
\dim_H(\Fre(\varrho,\delta,B)\cap H_\bom^J) \le \dim(H_\bom^J) -\kappa + \epsilon_1,
\]
where $\kappa=\dim(E^u_\bom)$.
\end{prop*}

The short derivation of \cite[Th.\,4.1]{BuSo21} from \cite[Prop.\,5.4]{BuSo21} in this form and the proof of the proposition do not require any changes. 

Incidentally, exactly the same mixed-up parameter dependence is present in \hbox{\cite[Prop.\,9.1]{BuSo18a}}, and the correction is also the same.
We would like to note that the proposition with the order of quantifiers as it was published, is, in fact, correct when the Rauzy induction is periodic; essentially, in the case of a single substitution, as it was shown in \cite{BuSol2}.

\subsubsection*{Acknowledgements}
We are grateful to Rodrigo Trevi\~no who pointed out the mistake to us.

\backmatter
\bibliographystyle{jepplain+eid}
\bibliography{bufetov-solomyak}
\end{document}

