We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-\varepsilon)|A|,\ \varepsilon>0$; namely, we show that $A$ is contained either in a one-dimensional coset progression of size comparable with $|A|$, or in a union of fewer than $\varepsilon^{-1}$ cosets of a finite subgroup. The bounds $3(1-\varepsilon)|A|$ and $\varepsilon^{-1}$ are best possible in the sense that none of them can be relaxed without tightening another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered as an extension of the latter onto arbitrary abelian groups, provided that there is “not too much torsion involved.”

Let $G$ be a finite abelian group of exponent $m\ge 2$. For subsets $A,S\subseteq G$, denote by $\partial_S(A)$ the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$. We show that if $S$ generates $G$, and $A$ is nonempty, then $\partial_S(A) \ge \frac{e}m\,|A|\ln\frac{|G|}{|A|}\, . $ Here the coefficient $e=2.718\ldots$ is best possible and cannot be replaced with a number larger than $e$. For homocyclic groups $G$ of exponent $m$, we find an explicit closed-form expression for $\partial_S(A)$ in the case where $S$ is the “standard" generating subset of $G$, and $A$ is an initial segment of $G$ with respect to the lexicographic order induced by $S$. Namely, we show that in this situation $ \partial_S(A) = |G|\,\omega_m(|A|/|G|), $ where $\omega_2$ is the Takagi function, and $\omega_m$ for $m\ge3$ is an appropriate generalization thereof. This particular case is of special interest, since for $m\in\{2,3,4\}$ it is known to yield the smallest possible value of $\partial_S(A)$, over all sets $A\subseteq G$ of given size. We give this classical result a new proof, somewhat different from the standard one. We also give a new, short proof of the Boros--Páles inequality $\omega_2(\frac{x+y}2) \le \frac{\omega_2(x)+\omega_2(y)}2 + \frac12\,|y-x|,$ establish an extremal characterization of the Takagi function as the (pointwise) maximal function, satisfying this inequality and the boundary condition $\max\{\omega_2(0),$ $\omega_2(1)\}\le 0$, and obtain similar results for the $3$-adic analogue $\omega_3$ of the Takagi function.

We show that, for a positive integer r, every minimal 1-saturating set in ${PG}(r-1,2)$ of size at least $\frac{11}{36}\,2^r+3$ either is a complete cap or can be obtained from a complete cap S by fixing some $s\in S$ and replacing every point $s'\in S\setminus\{s\}$ by the third point on the line through s and $s'$. Since, conversely, every set obtained in this way is a minimal 1-saturating set and the structure of large sum-free sets in an elementary abelian 2-group is known, this provides a complete description of large minimal 1-saturating sets. An algebraic restatement is as follows. Suppose that G is an elementary abelian 2-group and a subset $A\subseteq G\setminus\{0\}$ satisfies $A\cup2A=G$ and is minimal subject to this condition. If $|A|\ge\frac{11}{36}\,|G|+3$, then either A is a maximal sum-free set or there are a maximal sum-free set $S\subseteq G$ and an element $s\in S$ such that $A=\{s\}\cup\bigl(s+(S\setminus\{s\})\bigr)$. Our approach is based on characterizing those large sets A in elementary abelian 2-groups such that, for every proper subset B of A, the sumset $2B$ is a proper subset of $2A$.