Kemperman's inequality and Freiman's lemma via few translates

Let $G$ be a connected compact group equipped with the normalised Haar measure $\mu$. Our first result shows that given $\alpha ,\beta >0$, there is a constant $c=c(\alpha ,\beta )>0$ such that for any compact sets $A,B\subseteq G$ with $\alpha \mu (B)\ge \mu (A)\ge \mu (B)$ and $\mu (A)+\mu (B)\le 1-\beta$, there exist $b_1,\dots b_c\in B$ such that $$\mu (A\cdot \{b_1,\dots ,b_c\})\ge \mu (A)+\mu (B)\mathrm{.}$$ A special case of this, that is, when $G={\mathbb{T}}^d$, confirms a recent conjecture of Bollobás, Leader and Tiba. We also prove a quantitatively stronger version of such a result in the discrete setting of ${\mathbb{R}}^d$. Thus, given $d\in \mathbb{N}$, we show that there exists $c=c(d)>0$ such that for any finite, non-empty set $A\subseteq {\mathbb{R}}^d$ which is not contained in a translate of a hyperplane, one can find $a_1,\dots ,a_c\in A$ satisfying $$\mathrm{\mid }A+\{a_1,\dots ,a_c\}\mathrm{\mid }\ge (d+1)\mathrm{\mid }A\mathrm{\mid }-O_d(1)\mathrm{.}$$ The main term here is optimal and recovers the bounds given by Freiman’s lemma up to the $O_d(1)$ error term.