Geodesic
Power series and Peano curves
Izvestiya. Mathematics , Tome 27 (1986) no. 1, pp. 1-26.

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A series $\sum_{n=1}^\infty c_ne^{inx}$ with coefficients $\{c_n\}$ tending monotonically (decreasing) to zero is constructed whose sum $f(x)$ has the following property: for any complex number $$ w\in G=\biggl\{z:|z|\leqslant\frac32, \biggl|z-\frac32(-1+i)\biggr|\leqslant\frac{2.3}{\sqrt2}\biggr\} $$ the set $\{x\in(0,2\pi):f(x)=w\}$ has the cardinality of the continuum. Here the domain $G$ contains the segment $[-3/2,-1]$ on both the real and the imaginary axes. The construction is based on corresponding properties of lacunary trigonometric series, which are presented in detail. Bibliography: 8 titles. 
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A. S. Belov. Power series and Peano curves. Izvestiya. Mathematics , Tome 27 (1986) no. 1, pp. 1-26. http://geodesic.mathdoc.fr/item/IM2_1986_27_1_a0/

[1] Salem R., Zygmund A., “Lacunary power series and Peano curves”, Duke Math. J., 12:4 (1945), 569–578 | DOI | MR | Zbl

[2] Gaposhkin V. F., “O nulyakh trigonometricheskikh ryadov s monotonnymi koeffitsientami”, Izv. vyssh. uchebn. zavedenii, 1981, no. 9, 13–15 | MR | Zbl

[3] Dyachenko M. I., “O ryadakh Fure s monotonno ubyvayuschimi koeffitsientami i nekotorykh voprosakh gladkosti sopryazhennykh funktsii”, Soobsch. AN Gruz. SSR, 104:3 (1981), 533–536 | MR

[4] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, M., L., 1950

[5] Kahane J.-P., Weiss G., Weiss M., “On lacunary power series”, Arhiv for Matem., 5:1 (1963), 1–26 | DOI | MR | Zbl

[6] Weiss M., “Concerning a theorem of Paley on lacunary trigonometric series”, Acta Math., 102 (1959), 225–238 | DOI | MR | Zbl

[7] Belov A. S., “O summe lakunarnogo ryada”, Tr. Mosk. matem. ob-va, 33 (1975), 107–153 | MR | Zbl

[8] Belov A. S., “O neravenstve Sidona–Zigmunda v teorii lakunarnykh trigonometricheskikh ryadov”, Matem. zametki, 30:4 (1981), 501–515 | MR | Zbl