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Source. Jin's sixteen-page author manuscript, Theorem 7 on p. 12, proof pp. 12–14, and Corollary 2 on p. 14. Proposition 2 on p. 9 supplies the lower Banach density characterization.

For $C\subseteq\mathbb N_0$, use

$$\underline u(C)=\lim_{n\to\infty}\inf_{a\in\mathbb N_0} \frac{|C\cap[a,a+n]|}{n+1},\qquad \overline u(C)=\lim_{n\to\infty}\sup_{a\in\mathbb N_0} \frac{|C\cap[a,a+n]|}{n+1}.$$

Statement. For $A,B\subseteq\mathbb N_0$ and every integer $h\geq2$,

$$\underline u(A+B)\geq \underline u(A)^{1-1/h}\overline u(hB)^{1/h}.$$

The last factor is upper Banach density. The same formula holds for $h=1$ when $\underline u(A)>0$. At $h=1$ and $\underline u(A)=0$, only the trivial zero bound is recorded, rather than assigning $0^0=1$. There is no zero-membership or basis hypothesis on $B$, and $hB$ is an exact $h$-fold sum.

Proof sketch. Put $\alpha=\underline u(A)$ and $\beta=\overline u(hB)$. Proposition 2 says that $\underline u(C)\geq\rho>0$ if and only if for every $\varepsilon>0$ there is $N\geq0$ such that every integer interval $[a,b]\subseteq\mathbb N_0$ with $b-a\geq N$ has density greater than $\rho-\varepsilon$. This universal interval quantifier distinguishes the desired conclusion from Theorem 6.

For $0<\alpha<1$ and $\beta>0$, Jin fixes a long, suitably regular dense window of $hB$, using the selection argument from [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_6|Theorem 6]]. The argument must then work for an arbitrary sufficiently long interval of $A$. Combining the trimming idea of [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_4|Theorem 4]] with the block partition of Theorem 6, Jin deletes a terminal segment and thins each remaining full block separately to have density between $\alpha-\delta$ and $\alpha+\delta$. The blocks are long enough that the lower density bound and integer rounding permit this choice. The retained set has density at least $\alpha$ minus an error tending to zero with $\delta$ and the relative terminal loss. The selected window of $hB$ gives a disjoint block of image points for every occupied block of each nonempty subset of the retained set.

Applying the external [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_3|Theorem 3]] to the translated finite construction bounds the sumset density from below, up to losses tending to zero, by $\alpha^{1-1/h}\beta^{1/h}$. The proof handles the fixed translation and terminal losses uniformly over the starting point of the original long interval. Proposition 2 therefore gives a lower Banach density bound. The same construction covers $h=1$ when $\alpha>0$. If $\alpha=1$ and $\beta>0$, one fixed translate of $A$ already gives density one; if a factor vanishes with $h\geq2$, the asserted bound is zero.

Corollary 2. If $\overline u(hB)=1$, then

$$\underline u(A+B)\geq\underline u(A)^{1-1/h},\qquad \overline u(A+B)\geq\overline u(A)^{1-1/h},$$

using Theorems 7 and 6 respectively, with their order-one qualifications. Thus an upper Banach basis supplies both versions; lower Banach density of $hB$ need not equal one.

Coverage. This is a statement and proof sketch. The translated block construction, integer choices in each thinned block, and uniform endpoint estimates remain at the proof pointer pp. 12–14 and the claim on pp. 10–11. This page does not claim that these steps have received a full rewritten proof or an independent proof audit. They are not inputs to the complete Schnirelmann proof. The source's p. 13 also contains notation slips: the block count prints $|A_1\cap I_i|>\alpha-\delta$ without the factor $d_k$, and the displayed sum for the retained mass uses $k$ in a floor where the preceding block definition uses $d_k$; and the last line of the final display on p. 13 prints $\alpha-2d$ twice for $\alpha-2\delta$. The sketch uses block densities and the stated block width. It does not adopt the printed finite-loss calculation as a checked estimate.

Bears on. #35, as a distinct density analog. The mixed upper/lower hypothesis should not be substituted into that problem's Schnirelmann statement.