arc.

From , we have $$\label{rzg}
 \int_{{\mathbb{R}}/{\mathbb{Z}}\backslash E} 1_{F_\kappa} * 1_{[0,t]\text{ mod }{\mathbb{Z}}}\ d\mu_{{\mathbb{R}}/{\mathbb{Z}}} = o(1).$$ This bound can be used to show that partial sumsets of $F_\kappa$ and $[0,t] \text{ mod } {\mathbb{Z}}$ are mostly contained in $E$. However, it does not control the full sumset of these two sets. To get around this difficulty, we “smooth” $F_\kappa$ somewhat by replacing it with a modified set $H_\sigma$. More precisely, let $0 < \sigma < t$ be a small fixed quantity, and let $H_\sigma \subset {\mathbb{R}}/{\mathbb{Z}}$ be the set $H_\sigma \coloneqq F_\kappa +_{\sigma^2} ([0,\sigma] \text{ mod }{\mathbb{Z}})$. Observe that if $x \in H_\sigma$, then one has the pointwise lower bound $1_{F_\kappa} * 1_{[0,t]\text{ mod }{\mathbb{Z}}} \geq \sigma^2$ on the arc $x + ([0, t-\sigma] \text{ mod } {\mathbb{Z}})$; thus $$1_{F_\kappa} * 1_{[0,t]\text{ mod }{\mathbb{Z}}} \geq \sigma^2 1_{H_\sigma + [0, t-\sigma] \text{ mod } {\mathbb{Z}}}.$$ From this, and Markov’s inequality we conclude that all but $o(1)$ in measure of $H_\sigma + ([0, t-\sigma] \text{ mod } {\mathbb{Z}})$ lies in $E$. By , we conclude that $$\mu_{{\mathbb{R}}/{\mathbb{Z}}}( H_\sigma + ([0, t-\sigma] \text{ mod } {\mathbb{Z}})) \leq \mu_G(A) + t + o(1).$$ On the other hand, from Corollary \[kemp-cor\] and , one has 