TITLE: A sumset inequality
QUESTION [20 upvotes]: A friend asked me the following problem:

Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$

Here the notation $A+B$ denotes the set $\{ a+b\ : a\in A, b\in B\}$. 
It follows from Rusza's triangle inequality that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A-X|},$$ but since $|A-X|\geq |A|$, this isn't quite strong enough.
It is not hard to show that the inequality holds in either of the extremal cases where $A+A$ is minimal or maximal - that is when $|A+A|=|A|(|A|+1)/2$ or $|A+A|=2|A|-1$.
Is this statement true in general?
Rewriting the desired inequality as $$\frac{|A+X|}{|A+A|}\geq \frac{|X|}{|A|},$$ we are asking if adding $A$ to $X$ and looking at this as a subset of $A+A$ causes the proportion of elements to increase.
Edit: Numerical Calculations:
I did some numerical calculations where I let $A$ run through all possible subsets of $\{1,2,3,\dots,n\}$ and $X$ run through all proper subsets of $A$, and I calculated the ratio $$\frac{|A+X||A|}{|A+A||X|}.$$  The minimum of this ratio over all possible combinations of $A,X$ with $X\subsetneq A\subset\{1,2,3\dots,n\}$ appears in the following table: 
$$\begin{array}{cc}
\text{value of }n\ \  & \text{minimum}\\
3 & 4/3\\
4 & 6/5\\
5 & 8/7\\
6 & 10/9\\
7 & 12/11\\
8 & 14/13\\
9 & 16/15\\
10 & 18/17
\end{array} $$ Numerically the slightly stronger estimate $$\frac{|A+X|}{|A+A|}\geq\left(1+\frac{1}{2|X|+1}\right)\frac{|X|}{|A|}$$ seems to hold.

REPLY [7 votes]: In particular for $x = 5$, $K = 5$ and $M = 22$ in Sean Eberhard's example, the inequality is violated (assuming $[x]$ means $\lbrace 0,1,2,\ldots, x \rbrace$).  
I get $X = \{0,1,2,3,4,5,22,23,24,25,26,27,44,45,46,47,48,49,66,67,68,69,70,71,88,89,90,91,92
      ,93,110,111,112,113,114,115\}$
$A = \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,44,45,
 46,47,48,49,66,67,68,69,70,71,88,89,90,91,92,93,110,111,112,113,114,115,132,154,176,198,220,242,264,286,308,330,352,374,396,418,440,462,484,506,528,
      550\}$
and $|X| = 36$,
$|A| = 72$,
$|A+X| = 307$,
$|A+A| = 622$.