Suppose that $P$ is a set of $N$ points in the plane. Can we get a lower bound for the cardinality of the distance set $d(P)$ from the Szemerédi–Trotter theorem?


Here is my try.
The Szemerédi–Trotter theorem tells us that if $P$ is a set of $N$ points in the plane $\mathfrak{L}$ is a collection of $L$ lines in the plane then 
$$|I(P,\mathfrak{L})|\leq C(N^{2/3}L^{2/3}+N+L)$$
where $|I(P,\mathfrak{L})|=\{(p,\ell)\in P\times\mathfrak{L}:p\in\ell\}$ is the set of incidences and $C$ is some positive constant. 


Now I have already proven that this theorem also applies if we replace the collection of lines $\mathfrak{L}$ by a collection of circles in the plane $\mathfrak{C}$. 


In order to get a bound on $|d(P)|$, do the the following. 
Write $d(P)=\{d\_1,d\_2,\ldots,d\_m\}$.
For each point 
$p\_i\in P$ with $1\leq i\leq N$, let 
$\mathfrak{C}\_i$ be the collection of circles of center $p\_i$ and radii 
$d\_1,\ldots d\_m$. Let 
$$\mathfrak{C}=\bigcup\_{i=1}^N\mathfrak{C}\_i.$$
Now notice that $|\mathfrak{C}\_i|=|d(P)|$ so that $|\mathfrak{C}|=N|d(P)|$. 
Also notice that 
$$|I(P,\mathfrak{C\_i})|=N-1$$ since a point of $P$ can lie in at most one circle of $\mathfrak{C\_i}$. Futhermore, 
$$\bigcup\_{i=1}^NI(P,\mathfrak{C\_i})\subset I(P,\mathfrak{C})$$ so that
by the fact that the above union is disjoint and by Szemerédi–Trotter applied to $P$ and $\mathfrak{C}$ we have
$$N(N-1)\leq |I(P,\mathfrak{C})|\leq C(N^{2/3}|\mathfrak{C}|^{2/3}+|\mathfrak{C}|+N),$$
and therefore 
$$N(N-1)\leq C(N^{4/3}|d(P)|^{2/3}+N|d(P)|+N).$$


With the above inequality do I get a lower bound on $|d(P)|$? 



Lower bound on the distance set using incidences of points and circles
The Szemerédi–Trotter bound is known to be false for circles (it is true for circles with the same radii). There is a construction that gives $N^{2/3}|C|^{2/3}\log^{1/3}N$ incidences. The current best upper bound is about $N^{6/11}|C|^{9/11}$, and this is conjectured to be far from tight (see for example this [recent result of Sharir and Zahl](https://arxiv.org/abs/1604.07877)). You can find a similar correct argument in [Claim 1.6.4 here](http://math.caltech.edu/~2016-17/3term/ma191c-sec2/notes/000book.pdf). 


