Linear patterns in subset of the integers (for example, primes) such as arithmetical progressions is a hot topic in mathematics. Recently, much progress has been made in this area. For example, the next result of T. Sanders(in the paper "On Roth's theorem on progressions") gives a estimate of how large can a set be to make sure that it does not contain no non-trivial three-term arithmetic progressions.

Suppose that $A \subset \lbrace 1,\dots,N\rbrace$ contains no non-trivial three-term arithmetic progressions. Then \begin{equation*} |A| = O\left(\frac{N (\log \log N)^5 }{\log N}\right). \end{equation*}

My question is related to this result. For Roth's theorem we consider a linear equation. If we consider nonlinear equation, such as $x^2+y^2=z^2$ or $x^2-2y^2=1$, we can ask the same problems: Suppose that $A \subset \lbrace 1,\dots,N\rbrace$, if a nonlinear equation has no non-trivial solution in $A$, how large can it be. So my question is that whether there is any result about this problem.

Note: For equation $x^2+y^2=z^2$, it is not hard to prove that $A$ can have $N/2$ numbers. (Thank Mark Sapir for reminding me of this fact) So $|A|$ can be $O(N)$. But maybe we can consider this problem: for $b$ close to 1, if $N$ is sufficiently large and $|A|\geq bN$, can there be no non-trivial solution in $A$?

Thanks!