  We improve the result of our previous paper on translation invariant
quadratic forms in two special cases. We reduce the density bound
$|\mathcal{A}|/N = O((\log\log N)^{-c})$ to $|\mathcal{A}|/N = O((\log
N)^{-c})$ for most quadratic forms and handle almost diagonal equations in $s
\geq 6$ variables instead of $s \geq 8$.
