The Wiener norm of lifts of dense Sidon sets

We describe, with proof, an infinite class of examples of finite sets of integers $A$ for which $\mathrm{\parallel }\widehat{1_A}{\mathrm{\parallel }}_1\ge (c_{\mathcal{R}}-o_{\mathrm{\mid }A\mathrm{\mid }\to \mathrm{\infty }}(1))\sqrt{\mathrm{\mid }A\mathrm{\mid }}$ where $c_{\mathcal{R}}=0.953\dots$.