Injectivity of sums on two distinct rays ($|P+Q|=|P||Q|$)

Let finite sets $P,Q$ lie on two Euclidean rays with linearly independent direction vectors $u,v$. Write $p=su$, $q=tv$. If $p+q=p^{\prime }+q^{\prime }$, thenso linear independence forces $s=s^{\prime }$ and $t=t^{\prime }$. Hence the sumset map is injective andFor nonzero points on two distinct rays in the strictly positive quadrant, the directions are automatically linearly independent, and every sum lies in the open planar sector between them.

Merely distinct Euclidean rays are insufficient, because opposite rays have dependent directions. For $u\ne 0$, take $P=\{u,2u\}$ and $Q=\{-u,-2u\}$. The four ordered pairs give only three sums, $P+Q=\{-u,0,u\}$.