Lower bound on the distance set using incidences of points and circles

Question

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 $|d(P)|$?

Answer 1

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). You can find a similar correct argument in Claim 1.6.4 here.