Assume that the set $A$ does not have simple structures (such as all elements are odd numbers in $[1,M/2]$ etc., say we call them diophantine obstructions, as pointed out by @fedja).

What is the cardinality $n$ of a subset $A$ of $\{1,2,\ldots,M\}$ such that $(A+A) \cap A$ is empty and the set members don'tall obey simple diophantine obstructions.

Perhaps the following is easier? Instead of looking for $x+y=z$ for $x,y,z$ in $A,$ look for $x+y+z=0,$ which then is the question

Is the set $(A-A) \cap A$ nonempty?

where now one takes $A \subset \{-M,-M+1,\ldots,M\}$.