\[\begin{eqnarray}\sum_{n=0}^{\infty}(a_k\cos(2\pi n_k x)+b_k\sin(2\pi n_k x)),\quad\frac{n_{k+1}}{n_k}\geq \lambda >1\quad \text{for all}\quad k. \quad \quad (1)\end{eqnarray}\]
(It turns out that the sine and cosine form is more convenient than the complex exponential form in what follows). In other words, the frequencies of the trigonometric functions always grow by at least a factor of $\lambda$. It turns out that for such series, one can formulate some interesting convergence and integrability results that are not generally valid for trigonometric series. For example, Zygmund showed in 1930 that a lacunary series converges in a set of positive measure iff the sum of squares of its coefficients converges iff the lacunary series converges almost everywhere. This answers very broadly the question of convergence of such a series, while for a general trigonometric series the task of finding such a convergence criterion is hopeless. In the same paper, Zygmund proved that a lacunary series always converges in a dense set that even has the cardinality of the real numbers, unless the series is trivially divergent, that is $a_k^2+b_k^2\not \to 0$.
Another intriguing property of lacunary trigonometric series is that if such a series converges pointwise almost everywhere, its value distribution is random; in fact, the value distribution of its partial sums approaches the normal distribution. This was shown by Salem and Zygmund in 1947. It is also interesting to study whether a lacunary series is the Fourier series of some integrable function. Kolmogorov proved in 1924 that if this is the case, then this lacunary Fourier series converges almost everywhere, so for lacunary Fourier series, there are no problems with convergence (this of course follows from Carleson's result about pointwise almost everywhere convergence of $L^2$-functions, but the proof for lacunary series is much simpler, though not trivial).
It must also be mentioned that lacunary trigonometric series, whose frequencies grow too fast compared to the decay of the coefficients, are a good way to construct continuous nowhere differentiable functions, and indeed Weierstrass' example is a lacunary series.
The study of trigonometric series is also closely related to complex analysis, since the boundary values of an analytic function $f(z)=\sum_{n=0}^{\infty}a_nz^n$, defined in the unit disc, are given by the series $\sum_{n=0}^{\infty}a_ne^{in x}$, and conversely, a trigonometric series gives rise to an analytic function in the unit disc, whose boundary values are given by the trigonometric series, unless the coefficients grow too fast.
The above results (which are just a tiny proportion of what is known) should motivate the study of (lacunary) trigonometric series quite well. We will prove some of the mentioned results, following closely the original papers mentioned above.
