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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 $|\Delta(E)| = n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (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?

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?

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 $|\Delta(E)| = n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (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.

The best know results are due to Thomas Wolff ($d = 2$), and Burak Erdogan ($d \geq 3$) who have shown that the distance set of $E$ is of positive Lebesgue measure whenever.

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?

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 $|\Delta(E)| = n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (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?

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 $|\Delta(E)| = n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (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.

The best know results are due to Thomas Wolff ($d = 2$), and Burak Erdogan ($d \geq 3$) who have shown that the distance set of $E$ is of positive Lebesgue measure whenever.

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

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 $|\Delta(E)| = n^{\frac{2}{d}+ \underline{o}(1)}$ $\quad$ (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.

The best know results are due to Thomas Wolff ($d = 2$), and Burak Erdogan ($d \geq 3$) who have shown that the distance set of $E$ is of positive Lebesgue measure whenever.

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?