On lacunary series

This paper investigates properties of trigonometric $S_p$-systems ($p>2$) and Banach systems. In particular, the following theorems are established.
Theorem 1. {\it Let the system $\{\cos n_kx,\sin n_kx\}$ be an $S_p$-system $(n_k$ integers, $p>2)$. Then if the series $a_0+\sum a_k\cos n_kx+b_k\sin n_k x$ converges on a set of positive measure it follows that $a_0^2+\sum a_k^2+b_k^2<\infty$. If the same series converges to zero on a set of positive measure, all its coefficients are zero}.
Theorem 2. {\it Let the system $\{\cos n_kx,\sin n_kx\}$ be a Banach system. Let $\alpha(\{n_k\},[a,b])$ be the number of terms of the sequence $\{n_k\}$ that lie on $[a,b]$. Then}
$$\lim_{h\to+\infty}\sup_a\frac{\alpha(\{n_k\},[a,a+h])}h=0.$$

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