Geodesic
Power series and Peano curves
Izvestiya. Mathematics, Tome 27 (1986) no. 1, pp. 1-26 Cet article a éte moissonné depuis la source Math-Net.Ru

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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/

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