Additive energy of Piatetski-Shapiro sequences Let $c>1$, and let $A$ denote the set
$$
\Big\{ \lfloor n^c \rfloor, \quad 1 \leq n \leq N \Big\}.
$$ 
Thus $A$ consists of the first $N$ elements of a so-called Piatetski-Shapiro sequence.
The additive energy $E(A)$ of $A$ is defined as the number of solutions $(a_1,a_2,a_3,a_4) \in A^4$ of the equation $a_1 - a_2 = a_3 -a_4$.
Question: Is there an upper bound for $E(A)$ known? In particular, is it true that
$$
E(A) \ll N^{3 - \varepsilon}
$$
for some (small) $\varepsilon>0$, as $N \to \infty$?
(The case $c \geq 2$ is quite easy, but how about the other values of $c$ in the range $(1,2)$? Note that for $c < 2$ the set $A$ is not necessarily convex.)
 A: For $1<c<c_0$, for some constant $c_0$ the circle method will yield an asymptotic of size $$E(A)\sim \mathfrak{S}N^{4/c-1.}$$ where $\mathfrak{S}$ is a constant. In fact, when $c$ is not too large, one can even obtain an asymptotic for the additive energy of the Piatetski-Shapiro primes (see Balog and Friedlander). Letting $S_{A}(\theta)=\mathbb{E}_{n\leq N}1_{A}(n)e(n\theta)$ we may write $$E(A)=N^{3}\int_{0}^{1}|S_{A}(\theta)|^{4}d\theta.$$ The sum $S_A(\theta)$ can be handled by noting that $[(n+1)^{1/c}]-[n^{1/c}]$ is the indicator function for the Piatetski-Shapiro sequence, and then using the truncated Fourier series for the sawtooth function.
A: In "$L_1$-Norms of Exponential Sums and the Corresponding Additive Problem" Garaev and Kueh prove that if $F$ is in $C^3([1,N])$ with $F'(x)>0$, $F''(x)>0$ and $F'''(x)<0$, then the additive energy of $A=\{\lfloor F(1)\rfloor,\ldots,\lfloor F(N)\rfloor\}$ satisfies
$$ \frac{N^4}{F(N)+1}\ll E(A)\ll (F'(1)+1)N^{5/2} +\frac{N^2\log N}{F''(N)}.$$
In particular, if $F(x)=Ax^c$, with $A>0$ and $1< c\leq 3/2$, then
$$ N^{4-c}\ll_{c,A} E(A)\ll_{c,A} N^{4-c}\log N.$$
This paper builds on earlier work of Garaev, which presumably uses indicator trick mentioned in the first answer to bound the exponential sums, however the exponential sum estimates do not appear explicitly in the Garaev-Kueh paper.
PS: The ISSN for the Garaev-Kueh paper is 0232-2064.
PPS: I think the method mentioned by Eric might work for $1<c<2$, but I didn't write down all the details. You should be able to figure it out by staring at Heath-Brown's paper for a while (you should be able to avoid eq (5) if you're not summing over primes; proving (5) occupies the next 20 pages; in particular, you can re-apply Lemma 1 to the exponential sum at the end of the chain of inequalities on p. 247; overall I think the error term is $O(N^{1/2}(\log N)^b)$ for some power $b$.
