Lacunary Fourier series ($n_{j+1}/n_j\ge q>1$)

A Hadamard-lacunary Fourier series has nonzero frequencies supported on a positive sequence with the displayed gap ratio, possibly together with its negatives and finitely many extra modes. Such gaps strongly restrict additive frequency relations. They make moment estimates resemble those of independent random variables. For a ratio greater than three, the strongly lacunary polynomial L1-L2 bound follows directly by enumerating fourth-moment pairings.

Use normalized measure on the circle. If a trigonometric polynomial has frequency support $\{\pm n_j\}$ with $n_{j+1}>3n_j$, every zero sum of four signed frequencies pairs into opposite equal frequencies: the largest unpaired term would exceed the sum of the other three. Expanding the fourth moment and bounding its three possible pairings gives $\Vert p{\Vert }_4^4\le 3\Vert p{\Vert }_2^4$. The Holder inequality gives $\Vert p{\Vert }_2\le \Vert p{\Vert }_1^{1/3}\Vert p{\Vert }_4^{2/3}$ and hence the displayed estimate. Therefore an $L^1$ function with only these lacunary frequencies has square-summable coefficients: apply the estimate to its uniformly $L^1$-bounded Fejér sums and then pass to finite coefficient blocks. Finitely many extra modes do not affect the conclusion.