$$\mu_{{\mathbb{R}}/{\mathbb{Z}}}(H_\sigma) \geq \min( \mu_{{\mathbb{R}}/{\mathbb{Z}}}(F_\kappa) + t - \sigma, 1 ) - 2 \sigma \geq \mu_G(A) - 3 \sigma + o(1).$$ The situation here is reminiscent of that for which the inverse theorem for the Brunn-Minkowski inequality (see [@figalli], [@christ], [@christ2]), can be applied, but we are on the circle ${\mathbb{R}}/{\mathbb{Z}}$ instead of the line ${\mathbb{R}}$. However, as one of the sets involved is an arc, we can use the following elementary argument. As $H_\sigma$ is measurable, it is asymptotically equivalent to some finite union $K$ of arcs. For each $0 \leq s \leq t-\sigma$, the set $K_s \coloneqq K + ([0,s] \text{ mod } {\mathbb{Z}})$ is also a finite union of arcs, with $$\mu_G(A) - 3\sigma + o(1) \leq \mu_{{\mathbb{R}}/{\mathbb{Z}}}(K_0) \leq \mu_{{\mathbb{R}}/{\mathbb{Z}}}(K_{t-\sigma}) \leq \mu_G(A) + t + o(1).$$ It is easy to see that the function $s \mapsto \mu_G(K_s)$ is continuous and piecewise linear, with all slopes being positive integers. From the fundamental theorem of calculus, we thus see that the slope must in fact equal $1$ for all $s$ in $[0,t-\sigma]$ outside of a set of measure at most $4\sigma+o(1)$. The slope can only equal one when $K_s$ is an arc, thus $K_s$ must be an arc for some 