Arithmetic progressions in sumsets of geometric progressions

If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long’’ arithmetic progressions in the sumsets of the geometric progressions $1,a,a^2,a^3,\dots$ and $1,b,b^2,b^3,\dots$. Our proofs utilize recent applications of bounds for linear forms in logarithms to $S$-unit equations, and consequences of the modularity of Frey-Hellegouarch curves, together with elementary arguments.