Falconer distance problem in ℝ^2

Description of constant

The Falconer distance problem threshold $C_{34}=s_{\Delta }({\mathbb{R}}^2)$ in the plane is defined as $$s_{\Delta }({\mathbb{R}}^2)::=\inf \{s\in [0,2]:\forall \text{compact }E\subset {\mathbb{R}}^2,{\dim }_H(E)>s⟹|\Delta (E)|>0\}.$$ where for a compact set $E\subset {\mathbb{R}}^2$, the distance set is $$\Delta (E):=\{|x-y|:x,y\in E\}\subset [0,\infty )$$ <a href="#GIOW2018-def-distance-set">[GIOW2018-def-distance-set]</a>, ${\dim }_H$ denotes Hausdorff dimension, and $|\Delta (E)|$ denotes the 1-dimensional Lebesgue measure of $\Delta (E)$.

Known upper bounds

Known lower bounds

Additional comments and links

• The Falconer distance conjecture in the plane predicts that the lower bound of $1$ is sharp. <a href="#GIOW2018-conj-plane">[GIOW2018-conj-plane]</a> <a href="#GIOW2018-thm-5-4">[GIOW2018-thm-5-4]</a> <a href="#GIOW2018-lb-d-2">[GIOW2018-lb-d-2]</a>

References

• <a id="GIOW2018"></a>[GIOW2018] Guth, Larry; Iosevich, Alex; Ou, Yumeng; Wang, Hong. On Falconer’s distance set problem in the plane. Inventiones mathematicae 219 (3) (2020), 779–830. DOI: 10.1007/s00222-019-00922-7. Google Scholar. arXiv PDF
• <a id="GIOW2018-def-distance-set"></a>[GIOW2018-def-distance-set] loc: arXiv v1 PDF p.1, Introduction. quote: “For a set $E\subset {\mathbb{R}}^d$, define the distance set $$\begin{gathered} \Delta (E)= \\ |p-p^{\prime }|:p,p^{\prime }\in E \end{gathered}$$.”
• <a id="GIOW2018-conj-plane"></a>[GIOW2018-conj-plane] loc: arXiv v1 PDF p.1, Introduction. quote: “This led him to conjecture that if ${\dim }_H(E)>d/2$, then the Lebesgue measure of the distance set is positive. This is known as the Falconer Distance Conjecture.”
• <a id="GIOW2018-falconer-3-2"></a>[GIOW2018-falconer-3-2] loc: arXiv v1 PDF p.1, Introduction. quote: “He proved that if ${\dim }_H(E)>d+1/2$, then $L(\Delta (E))>0$.”
• <a id="GIOW2018-wolff-4-3"></a>[GIOW2018-wolff-4-3] loc: arXiv v1 PDF p.1, Introduction. quote: “In [37], Wolff proved that if $E\subset {\mathbb{R}}^2$ is a compact set with Hausdorff dimension greater than $4/3$, then $\Delta (E)$ has positive Lebesgue measure.”
• <a id="GIOW2018-thm-5-4"></a>[GIOW2018-thm-5-4] loc: arXiv v1 PDF p.1, Introduction. quote: “Theorem 1.1. If $E\subset {\mathbb{R}}^2$ is a compact set with Hausdorff dimension greater than $5/4$, then $\Delta (E)$ has positive Lebesgue measure.”
• <a id="GIOW2018-lb-d-2"></a>[GIOW2018-lb-d-2] loc: arXiv v1 PDF p.1, Introduction. quote: “Using an example based on the integer lattice, he showed for every $s\le d/2$ there exist sets of Hausdorff dimension $s$ for which $L(\Delta (E))=0$.”

• <a id="Fal1986"></a>[Fal1986] Falconer, K. J. On the Hausdorff dimensions of distance sets. Mathematika 32 (1985), no. 2, 206–212. DOI: 10.1112/S0025579300010998. Google Scholar.

• <a id="Wol1999"></a>[Wol1999] Wolff, Thomas. Decay of circular means of Fourier transforms of measures. International Mathematics Research Notices 1999 (10), 547–567. DOI: 10.1155/S1073792899000288. Google Scholar.

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.