Power series and Peano curves

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.