Hadamard-lacunary sets are Sidon

Set $Q=q^r$ and split the original indices of $\mathrm{\Lambda }$ into $r$ residue classes ${\mathrm{\Lambda }}_{\nu }$. Each class has successive ratios at least $Q\ge 3$. Set ${\epsilon }_{\lambda }=a_{\lambda }\mathrm{/}\mathrm{\mid }{a}_{\lambda }\mathrm{\mid }$ when $a_{\lambda }\mathrm{\ne }0$, and set it to $1$ otherwise. By [L1], the product $P_{\nu }={\prod }_{\lambda \in {\mathrm{\Lambda }}_{\nu }}(1+\mathrm{Re}({\epsilon }_{\lambda }{e}_{\lambda }))$ is nonnegative, has integral one, and has the prescribed coefficients on its class. Empty classes use the product $1$.

Every nonzero frequency $m$ in the product expansion is a signed sum of distinct frequencies of ${\mathrm{\Lambda }}_{\nu }$. If $L$ is its largest contributing frequency, the sum of the smaller frequencies is less than $L\mathrm{/}(Q-1)$, so $\mid \mathrm{\mid }m\mathrm{\mid }-L\mid <\frac{L}{Q-1}<(1-\frac{1}{q})L\mathrm{.}$ Every other frequency of the original sequence is at most $L\mathrm{/}q$ or at least $qL$; in the latter case its distance from $L$ is at least $(q-1)L>(1-1\mathrm{/}q)L$. Thus if $\mathrm{\mid }m\mathrm{\mid }$ belongs to the original sequence it must equal $L$, which lies in this class. In particular ${\widehat{P}}_{\nu }(-\lambda )=0$ for $\lambda \in \mathrm{\Lambda }\setminus {\mathrm{\Lambda }}_{\nu }$.

Finite character integration, [L1], and this vanishing give ${\int }_0^{1}F{P}_{\nu }=\frac{1}{2}{\sum }_{\lambda \in {\mathrm{\Lambda }}_{\nu }}\mathrm{\mid }{a}_{\lambda }\mathrm{\mid }\mathrm{.}$ For completeness, if $z=\int F{P}_{\nu }\mathrm{\ne }0$, multiplication by $\bar{z}\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ and taking real parts gives $\mathrm{\mid }z\mathrm{\mid }\le \int \mathrm{\mid }F\mathrm{\mid }{P}_{\nu }$; the same inequality is immediate if $z=0$. Apply [L2] to the real functions $\mathrm{\mid }F\mathrm{\mid }$ and $P_{\nu }$ to obtain $\frac{1}{2}{\sum }_{\lambda \in {\mathrm{\Lambda }}_{\nu }}\mathrm{\mid }{a}_{\lambda }\mathrm{\mid }\le {\int }_0^1\mathrm{\mid }F\mathrm{\mid }{P}_{\nu }\le \mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$

Summing over the $r$ classes yields ${\sum }_{\lambda \in \mathrm{\Lambda }}\mathrm{\mid }{a}_{\lambda }\mathrm{\mid }\le 2r\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}$. The constant depends only on $q$ and $r$, not on the finite support or coefficients, so this is the Sidon inequality.