Let $q$ be a positive integer. Is it true there exists a constant $C\_q$ such that the following inequality holds for any finite set $A$ of reals:
$$\displaystyle |A+qA|\ge (q+1)|A|-C\_q\qquad (1)$$
I got this idea from the well-known inequality $|A+A|\ge 2|A|-1$, so I was thinking about the general case, but no idea about it. I experimented a bit with small values of $|A|$ and it seems to be true. This paves the road for a far more general inequality 
$$ |kA+\ell A|\ge (k+\ell)|A|-C\_{k,\ell}\qquad (2)$$ with $k,\ell$ integers and $\text{gcd}(k,\ell)=1$. So, an answer would be welcome. A solution of $(1)$ is the thing I ask for, $(2)$ is just a bonus. And I haven't checked $(2)$ for small values yet. Anyways proving or disproving them both will be helpful. Thanks a lot. 



Sumsets and a bound
Inequality (2) is true provided $(k,\ell)=1$ - this is a recent result of Balog and Shakan in 'On the sum of dilates of a set', <http://arxiv.org/pdf/1311.0422.pdf>. 


They show that for any finite $A\subset\mathbb{Z}$ and integers $k,\ell$ such that $(k,\ell)=1$. 


$$ \lvert kA+\ell A\rvert \geq (k+\ell)\lvert A\rvert - (k\ell)^{(k+\ell-3)(k+\ell)+1}. $$


