The famous theorem of Roth (1953) states that if a set \(A\subseteq
\{1,\dots,N\}\) does not contain any non-trivial \(3\) term
arithmetic progression (a non-trivial solution to the equation
\(x+y=2z\)) then
\[
|A|\le C\frac{N}{\log\log N}
\]
for some absolute constant \(C\). This theorem became the focal
point of the additive combinatorics, and many attempts to improve
the above bound led to the development of interesting new
techniques. Roth's theorem was refined by Heath-Brown (1987),
Szemerédi (1990), Bourgain (1999, 2008), Sanders (2012) and very
recently by Bloom. The current best upper bounds, due to Sanders and
Bloom, give
\[
|A|\le \frac{N}{(\log N)^{1-o(1)}}.
\]
On the other hand, Behrend (1946) constructed subsets of
\(\{1,\dots,N\}\) of size
\[
e^{-C\sqrt{\log N}}N,
\] \(C>0,\) with no non-trivial progressions. This construction can
be easily extended to a wider class of equations, in particular to
equations of the form \(x_1+\dots+x_k=ky\), \(k\ge 2\).
We present the historical background of the problem, the techniques
developed in this area and new tight results concerning the equation
\(x+y+z=3w\).
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