We show that if
$A\subset \{1,\ldots ,N\}$
does not contain any nontrivial solutions to the equation
$x+y+z=3w$
, then
$$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$
where
$c>0$
is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent
$1/7$
cannot be replaced by any constant larger than
$1/2$
. We also establish a related result, which says that sumsets
$A+A+A$
contain long arithmetic progressions if
$A\subset \{1,\ldots ,N\}$
, or high-dimensional affine subspaces if
$A\subset \mathbb{F}_{q}^{n}$
, even if
$A$
has density of the shape above.