  We show that for every positive integer $k$ there are positive constants $C$
and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least
$C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$
contains a homogeneous $d$-dimensional generalized arithmetic progression of
size at least $c|A|^{d+1}$. This strengthens a result of Szemer\'edi and Vu,
who proved a similar statement without the homogeneity condition. As an
application, we make progress on the Erd\H{o}s--Straus non-averaging sets
problem, showing that every subset $A$ of $\{1, 2, \dots, n\}$ of size at least
$n^{\sqrt{2} - 1 + o(1)}$ contains an element which is the average of two or
more other elements of $A$. This gives the first polynomial improvement on a
result of Erd\H{o}s and S\'ark\"ozy from 1990.
