= Strongly lacunary polynomial L1-L2 bound
{title2=$\|p\|_2\leq\sqrt3\|p\|_1$}
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 $\|p\|_4^4\leq3\|p\|_2^4$. The <Holder inequality> gives $\|p\|_2\leq\|p\|_1^{1/3}\|p\|_4^{2/3}$ 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.
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