---

Source. Jin's sixteen-page author manuscript, Section 4, definition and Lemma 1 on p. 14, with the proof continued on p. 15. Page numbers are also PDF page numbers.

Write $C(a,b)=|C\cap[a,b]|$ for integer endpoints $a\leq b$.

Statement. Let $A,B\subseteq\mathbb N_0$, and let $h\geq1$ be an integer such that $hB=\mathbb N_0$. Suppose $0\leq a\leq b$ and

$$\gamma=\frac{A(a,b)}{b-a+1} =\min_{a\leq t\leq b}\frac{A(a,t)}{t-a+1}>0,$$

where the minimum is over integers (Jin's term for the equality is that $A$ has a minimal forward ratio on $[a,b]$). Then

$$(A+B)(a,b)\geq(b-a+1)\gamma^{1-1/h}.$$

For $h\geq2$ the assertion also holds when $\gamma=0$, with right side zero. For $h=1$ and $\gamma=0$, only the trivial nonnegative bound is asserted; the expression $0^0$ is not assigned a value.

Proof. First the minimal-prefix condition gives a bound for every tail. For $a<z\leq b$,

$$A(a,z-1)\geq\gamma(z-a),$$

so subtracting from $A(a,b)=\gamma(b-a+1)$ gives

$$A(z,b)\leq\gamma(b-z+1).$$

The same inequality is an equality when $z=a$.

Set $m=b-a$ and

$$F=(A\cap[a,b])-a\subseteq[0,m].$$

This is a nonempty finite set with $|F|=\gamma(m+1)$. Translation of the tail bound gives

$$|F\cap[z,m]|\leq\gamma(m-z+1)\qquad(0\leq z\leq m).$$

Take any nonempty $A'\subseteq F$ and put $z=\min A'$. Because $hB=\mathbb N_0$, all integers from $z$ to $m$ belong to $A'+hB$. No smaller integer belongs to this sumset, since its summands are nonnegative and every element of $A'$ is at least $z$. Therefore

$$(A'+hB)\cap[0,m]=[z,m],$$

where the right side denotes the integer interval. Also

$$0<|A'|\leq|F\cap[z,m]|\leq\gamma(m-z+1).$$

Consequently every subset in the minimum defining $D_{m,h}$ for $A_0=F$ satisfies

$$\frac{|(A'+hB)\cap[0,m]|}{|A'|} =\frac{m-z+1}{|A'|}\geq\frac1\gamma.$$

It follows that $D_{m,h}\geq\gamma^{-1}$. The external [[additive_bases/jin_2014_density_versions_plunnecke_inequality/theorem_3|Theorem 3]] now gives

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

Every element counted in $(F+B)\cap[0,m]$, after adding $a$, belongs to $(A+B)\cap[a,b]$. Multiplying the last display by $|F|=\gamma(m+1)$ proves

$$(A+B)(a,b)\geq(m+1)\gamma^{1-1/h}.$$

If $h\geq2$ and $\gamma=0$, nonnegativity proves the separately stated zero bound. This completes the proof.

Source normalization. The manuscript writes $A_0=A-a$ before invoking Theorem 3. The finite set $F$ above explicitly removes irrelevant elements outside $[a,b]$, ensuring that the translated set is nonnegative as that theorem requires. The printed sentence on p. 15 says $z=\min A_0$; the correct minimum in its subset argument is $\min A'$. The proof above checks the ratio for every nonempty $A'$, so neither a choice of the wrong minimum nor an unjustified equality of minima is needed. No zero is adjoined to the original set $A$.

Dependencies. Theorem 3 is an external Plünnecke graph inequality; the tail estimate and all other steps are included here.

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