Let $G$ be a connected compact group equipped with the normalised Haar measure $μ$. Our first result shows that given $α, β>0$, there is a constant $c = c(α,β)>0$ such that for any compact sets $A,B\subseteq G$ with $ αμ(B)\geqμ(A)\geq μ(B) $ and $ μ(A)+μ(B)\leq 1-β$, there exist $b_1,\dots b_c\in B$ such that \[ μ(A\cdot \{b_1,\dots,b_c\})\geq μ(A)+μ(B).\] 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 \[ |A+ \{a_1, \dots, a_c\}| \geq (d+1)|A| - O_d(1). \] The main term here is optimal and recovers the bounds given by Freiman's lemma up to the $O_d(1)$ error term.