Some refinements for translation invariant quadratic forms in dense sets

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 $s \geq 8$.