  We prove new lower bounds on the maximum size of sets $A\subseteq
\mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term
arithmetic progressions (consisting of three distinct points). More
specifically, we prove that for any fixed integer $m\ge 2$ and sufficiently
large $n$ (in terms of $m$), there exists a three-term progression free subset
$A\subseteq \mathbb{Z}_m^n$ of size $|A|\ge (cm)^n$ for some absolute constant
$c>1/2$. Such a bound for $c=1/2$ can be obtained with a classical construction
of Salem and Spencer from 1942, and improving upon this value of $1/2$ has been
a well-known open problem (our proof gives $c= 0.54$).
  Our construction relies on finding a subset $S\subset \mathbb{Z}_m^2$ of size
at least $(7/24)m^2$ with a certain type of reducibility property. This
property allows us to ``lift'' $S$ to a three-term progression free subset of
$\mathbb{Z}_m^n$ for large $n$ (even though the original set $S\subset
\mathbb{Z}_m^2$ does contain three-term arithmetic progressions).
