A subset $S\subseteq\mathbb{N}$ is said to be sum-free if whenever $s,t\in S$, then $s+t\notin S$. For instance the set of odd numbers is sum-free and has (lower and upper) asymptotic density 1/2.
Question. What is the supremum of lower asymptotic density that a set $S\subseteq\mathbb{N}$ can have being sum-free with respect to Knuth's approximate addition?
Let $a +_K b$ denote Knuth addition. It is easy to check that $a +_K b \equiv a + b$ mod $2$ (in fact mod $4$), so the odd numbers are Knuth-sum-free. On the other hand, note that if $a +_K b = a +_K b'$ then $b = b'$ (if $b \ne b'$ consider the least significant bit at which they differ). This property implies that any Knuth-sum-free set has upper density at most $1/2$. Indeed, if $A \subset \mathbb N$ is Knuth-sum-free and $a \in A$ then, for all $n$, $A \cap [n]$ and $a +_K (A \cap [n])$ are disjoint subsets of $[a + n]$ of cardinality $|A \cap [n]|$, so $|A \cap [n]| \le (a+n)/2$ and $\limsup |A \cap [n]|/n \le 1/2$.
