# Relationship between Erdos and Falconer distance problems

Given a set $E \subset \mathbb{R}^d$, define the distance set of $E$

$$\Delta(E) = \lbrace|x-y| : x,y \in E \rbrace,$$ where $|\cdot |$ is the usual Euclidean distance.

$\bullet$ The Erdos-distance conjecture roughly asserts that if a set $E \subset \mathbb{R}^d$ has finite cardinality $|E| = n$, then the distance set has size at least $|\Delta(E)| \ge n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (where $\underline{o}(1) \to 0$ as $n \to \infty$).

The case $d = 2$ was recently solved by Guth-Katz using an ingenious adaptation of Dvir's polynomial method along with the so-called algebraic method.

$\bullet$ The Falconer-distance conjecture is a continuous analogue of the Erdos-distance problem. It asserts that if the $E \subset \mathbb{R}^d$ has Hausdorff dimension $\dim_H(E) > \frac{d}{2}$ then the corresponding distance set $\Delta(E)$ has positive Lebesgue measure.

What is the relationship between these two results? It seems that the $\frac{2}{d}$ and the $\frac{d}{2}$ are related, but how, precisely? Is it possible to turn the Guth-Katz result into a result concerning the Falconer-distance problem?

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If $E \subset {\bf R}^d$ has bounded diameter and $\delta$-entropy n (i.e. it requires exactly $n$ balls of radius $\delta$ to cover $E$) for some $\delta>0$, then $\Delta(E)$ has $\delta$-entropy $\gg n^{\frac{2}{d}-o(1)}$.
Informally, this is a relaxation of the Erdos problem in which one now considers two numbers to be equivalent if they differ by $O(\delta)$. This makes it significantly harder to apply algebraic methods (which do not react well to this sort of "fuzzy" arithmetic, especially when dividing by small non-zero denominators), although there have been some partial successes in extending the algebraic method to this setting, such as Guth's proof of the endpoint multilinear Kakeya conjecture.