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 $x^2+y^2=pz^2$ 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}\$.