We study the Erdos distinct distance conjecture in the plane over an arbitrary field $\mathbb{F}$, proving that any $A$ with $|A|\leq\mathrm{char}(\mathbb{F})^{4/3}$ in positive characteristic, either determines $\gg |A|^{2/3}$ distinct pairwise distances from some point of $A$ to its other points, or the set $A$ lies on an isotropic line. We also establish for the special case of the prime residue field $\mathbb{F}_p$ that the condition $|A|\geq p^{5/4}$ suffices for $A$ to determine a positive proportion of the feasible $p$ distances. This significantly improves prior results on the problem.

On the Pinned Distances Problem over Finite Fields

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. 
We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to 
$$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. $$ The latter inequality is "threshold-breaking", for it shows for $\epsilon>0$, one has $$|AA| \le |A|^{1+\epsilon}\;\;\;\Rightarrow\;\;\; |A-A|\gg |A|^{\frac{3}{2}+c(\epsilon)},$$ with $c(\epsilon)>0$ if $\epsilon$ is sufficiently small. 
This implies that regardless of $\epsilon$, $$|AA-AA|\gg |A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$

Stronger sum--product inequalities for small sets

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. 
We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to 
$$|AA|^5 |A\pm A|^4 \gg |A|^{11-o(1)}\,,\;\;\;\;\mbox{hence} \;\; \;\;|AA|+|A\pm A|\gg |A|^{1+\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \gg |A|^{73-o(1)}. $$ The latter inequality is "threshold-breaking", for it shows for $\epsilon>0$, one has $$|AA| \le |A|^{1+\epsilon}\;\;\;\Rightarrow\;\;\; |A-A|\gg |A|^{\frac{3}{2}+c(\epsilon)},$$ with $c(\epsilon)>0$ if $\epsilon$ is sufficiently small. 
This implies that regardless of $\epsilon$, $$|AA-AA|\gg |A|^{\frac{3}{2}+\frac{1}{56}-o(1)}\,.$$

Stronger sum-product inequalities for small sets

Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type implications, e.g. that $A+A$ being small implies unbounded growth for a many enough times iterated product set $A \cdots A$.

Higher convexity and iterated sum sets

We study the Erdős–Falconer distance problem for a set A ⊂ F 2 $A\subset \mathbb {F}^2$ , where F $\mathbb {F}$ is a field of positive characteristic p $p$ . If F = F p $\mathbb {F}=\mathbb {F}_p$ and the cardinality | A | $|A|$ exceeds p 5 / 4 $p^{5/4}$ , we prove that A $A$ determines an asymptotically full proportion of the feasible p $p$ distances. For small sets A $A$ , namely when | A | ⩽ p 4 / 3 $|A|\leqslant p^{4/3}$ over any F $\mathbb {F}$ , we prove that either A $A$ determines ≫ | A | 2 / 3 $\gg |A|^{2/3}$ distances, or A $A$ lies on an isotropic line. For both large and small sets, the results proved are in fact for pinned distances. 

/pdf/on-the-pinned-distances-problem-in-positive-characteristic-2j9qaw1f.pdf

On the pinned distances problem in positive characteristic

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have 
$$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ 
This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to also prove the three-variable expansion result 
\[ 
|AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,. 
\] 
Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, that enables us to find positive proportion subsets with certain desirable properties.

On sum sets of convex functions.

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.

An Energy Bound in the Affine Group

We give a construction of a non-degenerate polynomial F ∈ R[x, y, z] and a set A of cardinality n such that |Z(F ) ∩ (A×A×A)| n 3 2 , thus providing a new lower bound construction for the Elekes–Szabó problem. We also give a related construction for the Elekes–Rónyai problem restricted to a subgraph. This consists of a polynomial f ∈ R[x, y] that is not additive or multiplicative, a set A of size n, and a subset P ⊂ A×A of size |P | n3/2 on which f takes only n distinct values. Mathematics Subject Classifications: 52C10

/pdf/constructions-for-the-elekes-szabo-and-elekes-ronyai-1394iz04mt.pdf

Constructions for the Elekes–Szabó and Elekes–Rónyai problems

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is $O(n^{11/6})$, improving a result of Chang and Solymosi. 
We also consider constructions for this problem. Alon, Ruzsa and Solymosi constructed an arrangement of four non-collinear $n$-pencils which determine $\Omega(n^{3/2})$ four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of $m$ $n$-pencils, whose centres are in general position, that determine $\Omega_m(n^{3/2})$ $m$-rich points.

Audie Warren

Papers

On sum sets of convex functions.

An Energy Bound in the Affine Group

An Energy Bound in the Affine Group

Constructions for the Elekes–Szabó and Elekes–Rónyai problems

Improved Bounds for Pencils of Lines