Geodesic
A question by Alexei Aleksandrov and logarithmic determinants
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2001) no. 3, pp. 308-317 
Mikhail Sodin. A question by Alexei Aleksandrov and logarithmic determinants. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2001) no. 3, pp. 308-317. http://geodesic.mathdoc.fr/item/JMAG_2001_8_3_a5/
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     author = {Mikhail Sodin},
     title = {A question by {Alexei} {Aleksandrov} and logarithmic determinants},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {308--317},
     year = {2001},
     volume = {8},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2001_8_3_a5/}
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We construct an analytic function $f$ of Smirnov's class in the unit disk such that $\mathrm{Re}\,f$ vanishes almost everywhere on the unit circle and $$ \liminf_{t\to\infty} t\operatorname{meas}\{\zeta:\,|\zeta|=1,\ |f(\zeta)|\ge t\}=0. $$ This answers negatively to the question posed by A. Aleksandrov. We also find new sufficient conditions for representations of functions of Smirnov's class by the Schwarz and Cauchy integrals. These conditions extend previous results by Aleksandrov.