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Source. Jin's sixteen-page author manuscript, Theorem 5 on p. 7. It explicitly refers to the earlier paper [8] for the proof.

Write $\overline d(C)=\limsup_{n\to\infty}|C\cap[1,n]|/n$ for upper asymptotic density.

Statement. There are $A,B\subseteq\mathbb N_0$ such that

$$\overline d(A)=\frac12,\qquad \overline d(2B)=1,\qquad \overline d(A+B)=\overline d(A)=\frac12.$$

Thus replacing every lower density in Theorem 4 by upper asymptotic density would be false: its right side for $h=2$ would be $1/\sqrt2>1/2$. The bar over the first $d(A)$ in the printed statement is upper density; no existence of the ordinary asymptotic density of $A$ is asserted here.

External proof pointer. Renling Jin, “Plünnecke's theorem for asymptotic densities,” Transactions of the American Mathematical Society 363 (2011), 5059–5070, DOI 10.1090/S0002-9947-2011-05533-9. The manuscript's reference [8] calls this “Plünnecke's Theorem for other densities.” Jin says the counterexample's proof there is already standard and does not repeat it. The earlier paper's construction and its exact internal locator have not been checked in this source unit.

Example 1 on the following manuscript page answers a different question: one cannot replace only $\underline d(hB)$ in Theorem 4 by $\overline d(hB)$. It is not the proof of the theorem stated above.

Bears on. #35, by identifying a limit of the density analogy. It does not contradict the Schnirelmann theorem.