Hi! Not sure if this is exactly what you are asking for, but for non-linear equations in three variables you can fix a $b<1$ arbitrarily close to $1$ and then construct an equation so that $|A| > bN$ and there are no solutions to the equation in $A$, by using congruence conditions as was done for $x^2+y^2=z^2$.

Take $x^2+y^2=pz^2$ for $p\equiv 3 \bmod 4$, $p$ sufficiently large, and form $A$ by deleting $p\mathbb{Z}$ from $\{1,\dots,N\}$. There are no solutions since $-1$ is not a square $\bmod p$.

But if $p$ is fixed, then one wonders how much larger $b$ can be beyond size $\frac{p-1}{p}$.

Hi! Not sure if this is exactly what you are asking for, but for non-linear equations in three variables you can fix a $b<1$ arbitrarily close to $1$ and then construct an equation so that $|A| > bN$ and there are no solutions to the equation in $A$, by using congruence conditions as was done for $x^2+y^2=z^2$.

Take $x^2+y^2=p^2z^2$ for $p\equiv 3 \bmod 4$, $p$ sufficiently large, and form $A$ by deleting $p\mathbb{Z}$ from $\{1,\dots,N\}$. There are no solutions since $-1$ is not a square $\bmod p$.

But if $p$ is fixed, then one wonders how much larger $b$ can be beyond size $\frac{p-1}{p}$.