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.
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1$\begingroup$ There is no way (2) could hold without at least requiring $k$ and $l$ to be coprime: if $m>1$ and $A=\{1,\dots,n\}$, then $|kmA+lmA|=|kA+lA|\le(k+l)n<(km+lm)n-C_{km,lm}$ for $n$ large enough. $\endgroup$– Emil JeřábekCommented Jun 15, 2014 at 15:43
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$\begingroup$ Thank you so I am adding the condition in the question. I knew the second one was faulty somehow. $\endgroup$– shadow10Commented Jun 15, 2014 at 15:46
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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 $(k,\ell)=1$.
$$ \lvert kA+\ell A\rvert \geq (k+\ell)\lvert A\rvert - (k\ell)^{(k+\ell-3)(k+\ell)+1}. $$
answered Jun 15, 2014 at 15:49