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For $g\in {\mathcal{H}}_N$, writeThe orthogonality of complex exponentials converts the linear configuration count intoExpand the difference between the products for $g_1$ and $g_2$ by changing one factor at a time. A typical term iswhere each $h_j$ is either $g_1$ or $g_2$. The assumed uniform norm bound controls the first factor by $\epsilon N$. The substitution $\alpha \mapsto k\alpha$ preserves an integral over the circle group, so Hölder's inequality and the three supplied $L^3$ bounds giveEach of the four terms is therefore $O(\epsilon N^3)$, and henceThis is Fourier stability of a linear configuration count.

The matrices are rotations about the third coordinate axis. Direct multiplication givesand $g(\theta {)}^{\mathsf{T}}g(\theta )=I$. Thus they form a one-parameter subgroup of the orthogonal group $O(3)$, isomorphic to the circle group.