Falconer's conjecture

In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact ${\displaystyle d}$-dimensional spaces. Intuitively, it states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. More precisely, if ${\displaystyle S}$ is a compact set of points in ${\displaystyle d}$-dimensional Euclidean space whose Hausdorff dimension is strictly greater than ${\displaystyle d/2}$, then the conjecture states that the set of distances between pairs of points in ${\displaystyle S}$ must have nonzero Lebesgue measure.

Formulation and motivation

Falconer (1985) proved that Borel sets with Hausdorff dimension greater than ${\displaystyle (d+1)/2}$ have distance sets with nonzero measure.[1] He motivated this result as a multidimensional generalization of the Steinhaus theorem, a previous result of Hugo Steinhaus proving that every set of real numbers with nonzero measure must have a difference set that contains an interval of the form ${\displaystyle (-\varepsilon ,\varepsilon )}$ for some ${\displaystyle \varepsilon >0}$.[2] It may also be seen as a continuous analogue of the Erdős distinct distances problem, which states that large finite sets of points must have large numbers of distinct distances.

Partial results

Erdoğan (2005) proved that compact sets of points whose Hausdorff dimension is greater than ${\displaystyle {\tfrac {d}{2}}+{\tfrac {1}{3}}}$ have distance sets with nonzero measure; for large values of ${\displaystyle d}$ this approximates the threshold on Hausdorff dimension given by the Falconer conjecture.[3]

For points in the Euclidean plane, a variant of Falconer's conjecture states that a compact set whose Hausdorff dimension is greater than or equal to one must have a distance set of Hausdorff dimension one. Falconer himself showed that this is true for compact sets with Hausdorff dimension at least 3/2, and subsequent results lowered this bound to 4/3.[4][5] It is also known that, for a compact planar set with Hausdorff dimension at least one, the distance set must have Hausdorff dimension at least 1/2.[6]

In 2018, Guth, Iosevich, Ou and Wang (arXiv:1808.09346) proved that if the Hausdorff dimension of a planar set is greater than 5/4, then there exists a point in the set such that the Lebesgue measure of the distances from the set to this point is positive.

Related conjectures

Proving a bound strictly greater than 1/2 for the dimension of the distance set in the case of compact planar sets with Hausdorff dimension at least one would be equivalent to resolving several other unsolved conjectures. These include a conjecture of Paul Erdős on the existence of Borel subrings of the real numbers with fractional Hausdorff dimension, and a variant of the Kakeya set problem on the Hausdorff dimension of sets such that, for every possible direction, there is a line segment whose intersection with the set has high Hausdorff dimension.[7]

Other distance functions

For non-Euclidean distance functions in the plane defined by polygonal norms, the analogue of the Falconer conjecture is false: there exist sets of Hausdorff dimension two whose distance sets have measure zero.[8][9]

References

1. Falconer, K. J. (1985), "On the Hausdorff dimensions of distance sets", Mathematika, 32 (2): 206–212 (1986), doi:10.1112/S0025579300010998, MR 0834490. See in particular the remarks following Corollary 2.3. Although this paper is widely cited as its origin, the Falconer conjecture itself does not appear in it.
2. Steinhaus, Hugo (1920), "Sur les distances des points dans les ensembles de mesure positive" (PDF), Fund. Math. (in French), 1: 93–104.
3. Erdoğan, M. Burak (2005), "A bilinear Fourier extension theorem and applications to the distance set problem", International Mathematics Research Notices, 23: 1411–1425, CiteSeerX 10.1.1.121.7673, doi:10.1155/IMRN.2005.1411.
4. Bourgain, Jean (1994), "Hausdorff dimension and distance sets", Israel Journal of Mathematics, 87 (1–3): 193–201, doi:10.1007/BF02772994, MR 1286826.
5. Wolff, Thomas (1999), "Decay of circular means of Fourier transforms of measures", International Mathematics Research Notices (10): 547–567, doi:10.1155/S1073792899000288, MR 1692851.
6. Mattila, Pertti (1987), "Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets", Mathematika, 34 (2): 207–228, doi:10.1112/S0025579300013462, MR 0933500.
7. Katz, Nets Hawk; Tao, Terence (2001), "Some connections between Falconer's distance set conjecture and sets of Furstenburg type", New York Journal of Mathematics, 7: 149–187, MR 1856956.
8. Falconer, K. J. (May 2004), "Dimensions of intersections and distance sets for polyhedral norms", Real Analysis Exchange, 30 (2): 719–726, MR 2177429.
9. Konyagin, Sergei; Łaba, Izabella (2006), "Distance sets of well-distributed planar sets for polygonal norms", Israel Journal of Mathematics, 152: 157–179, arXiv:math/0405017, doi:10.1007/BF02771981, MR 2214458.