A huge part of the investigation in the area of additive combinatorics asks for the answer of the following question: given an arithmetic pattern (for instance, $x+y=2z$, or $x+y=z+t$, or $x+y=z$), give the largest size of a subset $A\subseteq [n]$ avoiding the considered configuration. For instance, when dealing with $x+y=2z$ we have the theory starting from Roth's theorem and whose generalization, Szemerédi's theorem, has been a key and very influential result on the area.
My question is the following: which results are known concerning lower bounds for this type of problems when taking subsets of the set of non-negative integers instead of the finite interval?
For instance, in the case of the Sidon equation $x+y=z+t$, the upper and the lower bound ($\sqrt{n}$, up to the error matches): there are explicit constructions reaching the square root value, and it is possible to show that the upper bound must be of the same order. However, concerning lower bounds, in the infinite case the only thing known is that there exists Sidon sets $A$, such that $|A(n)|=|\{a\in A: a\leq n\}|=n^{\sqrt{2}-1+o(1)}$ (this is a very clever result first by Ruzsa, and then by Cilleruelo).
For instance, what is it the best known in the case of the 3-AP equation? In the finite case we have Behrend construction, but I guess (as in the case of Sidon) that this cannot be lifted to an infinite family. I know that the integers which uses 0,1 in their ternary representation defines a 3-AP-free set (hence, density $n^{2/3}$, but not sure if something better is known.
So, in general, what is known for explicit dense infinite families avoiding a given arithmetic pattern?