[Paper Review] A discretised projection theorem in the plane

Tuomas Orponen|arXiv (Cornell University)|Jul 24, 2014

Advanced Topology and Set Theory8 references3 citations

TL;DR

This paper establishes a discretised projection theorem in the plane, showing that for any $1/2 \leq s < 2 - \sqrt{2}$, there exists $\sigma(s) < s$ such that if $A \subset [0,1]$ is a $(\delta,1/2)$-set of size $\sim \delta^{-1/2}$ and $E \subset [0,1]$ contains $\gtrsim \delta^{-\sigma}$ $\delta^s$-separated points, then some translate $t \in E$ ensures $A + tA$ contains $\gtrsim \delta^{-s}$ $\delta$-separated points. The proof combines structural analysis of sets near-optimal for Kaufman's projection bound with an adapted version of Solymosi's sum-product argument.

ABSTRACT

The main result of this paper is that for any $1/2 \leq s &lt; 2 - \sqrt{2} \approx 0.5858$, there is a number $σ= σ(s) &lt; s$ with the following property. Let $δ&gt; 0$ be small, assume that $A \subset [0,1]$ is a $(δ,1/2)$-set, and that $E \subset [0,1]$ contains $\gtrsim δ^{-σ}$ roughly $δ^{s}$-separated points. Then there exists a number $t \in E$ such that $A + tA$ contains $\gtrsim δ^{-s}$ $δ$-separated points. For $σ= s$, this is essentially a consequence of Kaufman's well-known bound for exceptional sets of projections. Our proof consists of a structural observation concerning sets, for which Kaufman's bound is near-optimal, combined with (an adaptation of) Solymosi's argument for his "$4/3$" sum-product theorem.

Motivation & Objective

• To establish a quantitative strengthening of Kaufman’s exceptional set bound for projections in the discretised setting.
• To address the gap in the continuous projection problem where the sharp conjecture $\dim\{e : \dim\pi_e(K) < s\} \leq 2s - 1$ remains unproven.
• To provide a concrete, effective improvement over Kaufman’s bound by introducing a smaller exponent $\sigma(s) < s$ in the size of the set $E$.
• To demonstrate that the structural properties of sets nearly extremal for Kaufman’s bound allow for stronger sum-product type conclusions via adapted discrete incidence geometry.

Proposed method

• Define a $(\delta,s)$-set as a $\delta$-separated set with controlled doubling on balls: $|A \cap B(x,r)| \lesssim (r/\delta)^s$.
• Use the assumption that $A$ is a $(\delta,1/2)$-set of size $\sim \delta^{-1/2}$ to model a set of Hausdorff dimension $1/2$.
• Assume for contradiction that no such $t \in E$ exists with $|A + tA| \gtrsim \delta^{-s}$, implying $A \times A$ is nearly extremal for Kaufman’s bound.
• Identify a 'fan' structure in $A \times A$, where many points lie on few $\delta$-tubes from a common origin, based on near-optimality of Kaufman’s bound.
• Adapt Solymosi’s sum-product argument to show that such fan structures force $|A + A| \gtrsim \delta^{-1/2 - c}$ for some $c > 0$, contradicting the existence of $1 \in E$ if $s < 1/2 + c$.
• Use the Szemerédi-Trotter incidence theorem to bound incidences between lines and points in the fan configuration, leading to a contradiction when $s < 2 - \sqrt{2}$.

Experimental results

Research questions

• RQ1 Can a quantitative improvement over Kaufman’s exceptional set bound be achieved in the discretised setting for $s < 2 - \sqrt{2}$?
• RQ2 What structural properties emerge in sets $A$ for which Kaufman’s bound is nearly sharp?
• RQ3 Can Solymosi’s sum-product method be adapted to the discretised projection setting to derive stronger size estimates for $A + tA$?
• RQ4 Is it possible to find $\sigma(s) < s$ such that $E$ of size $\delta^{-\sigma}$ guarantees $A + tA$ has $\delta^{-s}$ $\delta$-separated points for some $t \in E$?
• RQ5 How do non-concentration assumptions in the discretised setting affect the applicability of incidence geometry tools like Szemerédi-Trotter?

Key findings

• For any $s \in [1/2, 2 - \sqrt{2})$, there exists $\sigma(s) < s$ such that if $A$ is a $(\delta,1/2)$-set of size $\sim \delta^{-1/2}$ and $E$ contains $\gtrsim \delta^{-\sigma}$ $\delta^s$-separated points, then $A + tA$ contains $\gtrsim \delta^{-s}$ $\delta$-separated points for some $t \in E$.
• The value of $\sigma(s)$ is strictly less than $s$, providing a non-trivial improvement over the standard application of Kaufman’s bound, which would give $\sigma = s$ only with logarithmic losses.
• The proof identifies a 'fan' structure in $A \times A$ when the set is nearly extremal for Kaufman’s bound, where many points lie on few $\delta$-tubes from a common origin.
• By adapting Solymosi’s sum-product argument to this fan configuration, the method shows $|A + A| \gtrsim \delta^{-1/2 - c}$ for some $c > 0$, contradicting the existence of $1 \in E$ if $s < 1/2 + c$.
• The contradiction is derived using the Szemerédi-Trotter incidence theorem, which bounds the number of incidences between lines and points in the fan, leading to a lower bound on $|A + A|$ that forces $s < 2 - \sqrt{2}$.
• The result is sharp in the sense that $2 - \sqrt{2} \approx 0.5858$ is the threshold beyond which the argument fails, as shown by the example in the appendix.