Small complete 3-term progression free sets in cyclic groups and vector spaces

A classical extremal problem on progression free sets is to determine the maximum size of a $3$-term arithmetic progression free set in algebraic structures, for instance in intervals of integers or in finite vector spaces. To determine the minimum size of a complete $3$-term arithmetic progression free set is a lower-end analogue of this problem. It is also closely related to complete caps and saturating sets in finite geometry. A simple counting argument shows that the order of magnitude of the minimum size is at least the square root of the cardinality of the structure. Addressing two open problems, we show that this lower bound is essentially tight. First, for every cyclic group $\mathbb{Z}_m$, we give explicit constructions of complete $3$-AP-free sets whose size is less than $2\sqrt m$. For $m\ge81$ the constructed sets satisfy the stronger, so-called complete $(2,-1)$-avoiding property; the remaining cases $m<81$ are covered by a finite verification. Second, we resolve the vector space variant in a weaker sense by showing that for every fixed odd prime $p$ and $\varepsilon>0$, there is a constant $C_{p, \varepsilon}$ such that $$a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)\le C_{p, \varepsilon}\,n^{1+\varepsilon}\,p^{n/2} =p^{n/2+o(n)}$$ holds for the minimum size $a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)$ of a complete 3-AP-free subset of $\mathbb{F}_p^n$, for all $n\ge1$.