Sumset growth in progression-free sets

We study the growth of sumsets $\mathcal {A}+\mathcal {B}\subset \mathcal {S}\subset G$, where $\mathcal {S}$ does not contain an arithmetic progression of length $2k+1$, and where $G$ is a commutative group, in which every nonzero element has an order of at least $2k+1$. More specifically, we show the following: if $\mathcal {A},\mathcal {B} \subset G$ are sets such that $\mathcal {A}+\mathcal {B}$ does not contain an arithmetic progression of length $2k+1$, then \[ |\mathcal {A}+\mathcal {B}| \geq |\mathcal {A}|^{\frac {2k-1}{3k-2}} |\mathcal {B}|^{\frac{k}{3k-2}} . \]

As an application we derive upper bounds on the cardinality of the summands in sumsets $\mathcal {A} + \mathcal {B} + \mathcal {C}$ contained in the set of $t$-th powers, where $t \geq 2$ is an integer. In particular, we show that $\min (|\mathcal {A}|,|\mathcal {B}|,|\mathcal {C}|)\ll (\log N)^{4/5}$ for $t=2$, and $\min (|\mathcal {A}|,|\mathcal {B}|,|\mathcal {C}|)\ll _t (\log N)^{1/2}$ for $t \geq 3$.