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For the physical-space L2 norm use $\Vert h{\Vert }_2^2={\mathbb{E}}_x|h(x){|}^2$. By part (i) and the Parseval identity on a finite group,Split this sum into the large spectrum $K$ and its complement. Because $u$ belongs to the Bohr set in the question, for $r\in K$,Consequently the contribution from $K$ is at most ${\epsilon }^2{\sum }_r|\hat{a}(r){|}^4\le {\epsilon }^2{\alpha }^3$ by part (ii). Outside $K$, $|\hat{a}(r)|<\theta$, and $|1-{\omega }^{-ru}|\le 2$. Thus that contribution is at mostAdding the estimates provesThe argument also covers empty $A$ or empty $K$. As usual the radius and threshold are nonnegative; a negative radius makes the premise empty whenever $K$ is nonempty. The estimate expresses Bohr-set almost periodicity of a convolution: a translation of a function by an element of the Bohr set barely changes the large Fourier coefficients on a finite abelian group, while the small ones have little total energy.