The following problem is somehow hidden in [this recently asked question](https://mathoverflow.net/questions/204020/is-the-set-aaa-always-at-least-as-large-as-aa), but I believe that it deserves to be asked explicitly.



> 
> Is it true that for any finite set $A$ of real numbers, and any real $\lambda\notin\{0,-1\}$, one has
>  $$ |A+\lambda A| \ge |A+A| $$
>  (where $\lambda A=\{\lambda a\colon a\in A\}$ is the dilate of $A$ by the factor $\lambda$, and $A+B=\{a+b\colon a\in A,\ b\in B\}$ is the sumset of $A$ and $B$)?
> 
> 
> 


The energy version seems equally interesting to me. Let $T\_A(\lambda)$ denote the number of representations of $\lambda$ in the form $\frac{a\_1-a\_2}{a\_3-a\_4}$ with $a\_1,a\_2,a\_3,a\_4\in A$. It is easily seen that $T\_A(-\lambda)=T\_A(\lambda)$ and $T\_A(\lambda)<T\_A(0)=|A|^2(|A|-1)$ for any real $\lambda\ne 0$.



> 
> Is it true that for any finite set $A$ of real numbers, and any real $\lambda\ne 0$, one has
>  $$ T\_A(\lambda) \le T\_A(1) ? $$
> 
> 
> 


I have not done any computations, so maybe it is possible to find a counterexample just by a computer search.




---


Both questions have now received nice and exhaustive answers thanks to Boris Bukh, Kevin Costello, and Terry Tao (who has actually answered even before the question got asked). Unfortunately, I cannot accept more than one answer; so, I am accepting only that which, for some reason, got less votes.



Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold?
It seems the energy version is true if make the additional assumption that $\lambda=c/d$ is rational, meaning that $T\_A(\lambda)$ counts the number of solutions in $A$ to
$$d(a\_1-a\_2)-c(a\_3-a\_4)=0 \, \, \, \, \, \, (\*).$$ Using a Freiman isomorphism together with translation invariance, we can assume that $A$ is a subset of integers lying in $[0,n]$.


Let $p$ be a prime larger than $2(c+d)n$. We now think of $A$ as a subset of $Z\_p$, which does not change the solution count. Let $f$ be the characteristic function of $A$, and $\hat{f}$ its Fourier transform. The number of solutions to $(\*)$ is then given by
$$\sum\_{\xi \in Z\_p} \hat{f}(d \xi) \hat {f} (-d \xi) \hat{f}(c \xi) \hat{f}(-c \xi)=\sum\_{\xi \in Z\_p} |\hat{f} (d \xi)|^2 |\hat{f} (c \xi)|^2$$


Since $c$ and $d$ are nonzero, the two sequences $|\hat{f} (d \xi)|^2$ and $|\hat{f} (c \xi)|^2$ are permutations of each other. So it follows from the [rearrangement inequality](http://en.wikipedia.org/wiki/Rearrangement_inequality) that the sum is maximized when $c=d$, which corresponds to $\lambda=1$.


It seems intuitive that moving from the rational to the reals can't increase the number of solutions, but I don't see an obvious way of extending this.


