  We prove results on the structure of a subset of the circle group having
positive inner Haar measure and doubling constant close to the minimum. These
results go toward a continuous analogue in the circle of Freiman's $3k-4$
theorem from the integer setting. An analogue of this theorem in $\mathbb{Z}_p$
has been pursued extensively, and we use some recent results in this direction.
For instance, obtaining a continuous analogue of a result of Serra and Z\'emor,
we prove that if a subset $A$ of the circle is not too large and has doubling
constant at most $2+\varepsilon$ with $\varepsilon<10^{-4}$, then for some
integer $n>0$ the dilate $n\cdot A$ is included in an interval in which it has
density at least $1/(1+\varepsilon)$. Our arguments yield other variants of
this result as well, notably a version for two sets which makes progress toward
a conjecture of Bilu. We include two applications of these results. The first
is a new upper bound on the size of $k$-sum-free sets in the circle and in
$\mathbb{Z}_p$. The second gives structural information on subsets of
$\mathbb{R}$ of doubling constant at most $3+\varepsilon$.
