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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 $b\in (1/2,1)$, b$close to 1, if$N$is sufficiently large and$|A|\geq bN$, can there be no non-trivial solution in$A$? Thanks! 4 added 47 characters in body Linear patterns in subset of the integers (for example, primes) such as arithmetical progressions is a hot topic in mathematics. Recently, much progress has been made in this area. For example, the next result of T. Sanders(in the paper "On Roth's theorem on progressions") gives a estimate of how large can a set be to make sure that it does not contain no non-trivial three-term arithmetic progressions. Suppose that$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\in (1/2,1)$, if$N$is sufficiently large and$|A|\geq bN$, can there be no non-trivial solution in$A$? Thanks! 3 added 269 characters in body Linear patterns in subset of the integers (for example, primes) such as arithmetical progressions is a hot topic in mathematics. Recently, much progress has been made in this area. For example, the next result of T. Sanders(in the paper "On Roth's theorem on progressions") gives a estimate of how large can a set be to make sure that it does not contain no non-trivial three-term arithmetic progressions. Suppose that$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$numbers. So$|A|$can be$O(N)$. But maybe we can consider this problem: for$b\in (1/2,1)$, if$N$is sufficiently large and$|A|\geq bN$, can there be no non-trivial solution in$A\$?

Thanks!

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