The logarithmic lower-bound conjecture for Fourier norms on compact Abelian groups

Let $G$ be a compact Abelian group with dual group $\widehat G$. Let $A\subset G$ have density $\alpha$, and let $M$ be a positive parameter. For a finite subgroup $V\leq\widehat G$, write $\{x\}$ for the fractional part of $x$.

Logarithmic Fourier-norm conjecture. If, for every finite subgroup $V\leq\widehat G$ with $|V|\leq M$, one has

$$\{\alpha|V|\}\bigl(1-\{\alpha|V|\}\bigr)\gg1,$$

then

$$\|\chi_A\|_{A(G)}\gg\log M.$$

The conjecture seeks the optimal general lower bound in the question posed by the paper. The paper proves weaker bounds, including $\log\log M$ for compact vector spaces over $\mathbb F_2$ and $\log\log\log M$ in the general compact Abelian setting; the conjectured logarithmic bound remains open in the supplied context.