Lacunary series

Remarks

Example: $\sum_{k \ge 0} 2^{-k} \cos(2^k x)$ is lacunary with $q = 2$, since $2^{k+1}/2^k = 2$.

Non-example: $\sum_{k \ge 1} \frac{\cos(k^2 x)}{k^2}$ has frequencies $n_k = k^2$ with $n_{k+1}/n_k = (1 + 1/k)^2 \to 1$. So no $q > 1$ is a lower bound for all ratios, and the series is not lacunary in the sense of Hadamard.

A theorem of Sidon says that if a lacunary trigonometric series is the Fourier series of a bounded function, then $\sum_k (\abs{a_k} + \abs{b_k}) < \infty$. For power series the Hadamard gap theorem applies. Weaker gap conditions, such as $n_k / k \to \infty$ in the Fabry gap theorem, are also studied.