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Source. Jin's sixteen-page author manuscript, Theorem 3 on p. 3 and the proof references on p. 4. The source heading attributes the theorem to “Plünnecke, 1957.” This page records the statement used by Jin, without independently establishing that historical date.

Statement. Let $A_0,B\subseteq\mathbb N_0$, let $n\geq0$ and $h\geq1$ be integers, and suppose $A_0\cap[0,n]\ne\varnothing$. For $1\leq i\leq h$ define

$$D_{n,i}=\min_{\varnothing\ne A'\subseteq A_0\cap[0,n]} \frac{|(A'+iB)\cap[0,n]|}{|A'|}.$$

Then

$$D_{n,1}\geq D_{n,2}^{1/2}\geq\cdots\geq D_{n,h}^{1/h}.$$

The minimum is over a nonempty finite family of nonempty finite sets, so it is attained and every denominator is positive. The sets $A_0$ and $B$ need not be finite; all counted images lie in the finite interval $[0,n]$. Neither $0\in B$ nor a basis hypothesis is required for this theorem. For $h=1$ the displayed assertion has only its first term. The height-zero case has no ratios and is not used.

External dependency and proof pointer. Jin supplies no proof here. On p. 4 Jin identifies this as a consequence of the general graph inequality and points to:

• M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, 1996, Chapter 7 (Jin's reference [9]).
• I. Z. Ruzsa, “Sumsets and Structure,” in Combinatorial Number Theory and Additive Group Theory, Birkhäuser, 2009, 87–210, Chapter 1 (reference [11]).

These are precise external proof pointers. Their graph proofs have not been compiled or independently checked in this source unit. In particular, this page is not a claim that the proof in Plünnecke's unacquired 1970 article has been read.

Use in the density proof. For any nonempty $F\subseteq[0,n]$, applying the theorem with $A_0=F$ gives

$$\frac{|(F+B)\cap[0,n]|}{|F|} \geq D_{n,1}\geq D_{n,h}^{1/h}.$$

Thus a lower bound for the $h$-fold image ratio of every nonempty subset of $F$ yields a lower bound for the one-fold image of $F$ itself. This is the only graph-theoretic input in [[additive_bases/jin_2014_density_versions_plunnecke_inequality/lemma_1|Lemma 1]].

Bears on. #35, through [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_2|Theorem 2]].