geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...

, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...

s between points in

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

d

-dimensional spaces. Intuitively, it states that a set of points that is large in its

Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...

must determine a set of distances that is large in

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

. More precisely, if

S

is a compact set of points in

d

-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

whose Hausdorff dimension is strictly greater than

d/2

, then the

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

states that the set of distances between pairs of points in

S

must have nonzero

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

Formulation and motivation

proved that

Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...

s with Hausdorff dimension greater than

(d+1)/2

have distance sets with nonzero measure. He motivated this result as a multidimensional generalization of the

Steinhaus theorem In the mathematical field of real analysis, the Steinhaus theorem states that the difference set of a set of positive measure contains an open neighbourhood of zero. It was first proved by Hugo Steinhaus. Statement Let ''A'' be a Lebesgue-measura ...

, a previous result of

Hugo Steinhaus Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Unive ...

proving that every set of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s with nonzero measure must have a

difference set In combinatorics, a (v,k,\lambda) difference set is a subset D of size k of a group G of order v such that every nonidentity element of G can be expressed as a product d_1d_2^ of elements of D in exactly \lambda ways. A difference set D is said ...

that contains an interval of the form

(-\varepsilon,\varepsilon)

for some

\varepsilon>0

. It may also be seen as a continuous analogue of the

Erdős distinct distances problem In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 and almost proven by Larry Guth and Nets Katz in 2015. ...

, which states that large finite sets of points must have large numbers of distinct distances.

Partial results

proved that compact sets of points whose Hausdorff dimension is greater than

\tfrac + \tfrac

have distance sets with nonzero measure; for large values of

d

this approximates the threshold on Hausdorff dimension given by the Falconer conjecture. For points in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

, Borel sets of Hausdorff dimension greater than 5/4 have distance sets with nonzero measure and, more strongly, they have a point such that the Lebesgue measure of the distances from the set to this point is positive. A variant of Falconer's conjecture states that, for points in the plane, a compact set whose Hausdorff dimension is greater than or equal to one must have a distance set of Hausdorff dimension one. This follows from the results on measure for sets of Hausdorff dimension greater than 5/4. For a compact planar set with Hausdorff dimension at least one, the distance set must have Hausdorff dimension at least 1/2.

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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

on the existence of

Borel Borel may refer to: People * Borel (author), 18th-century French playwright * Jacques Brunius, Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding ...

subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...

s of the real numbers with fractional Hausdorff dimension, and a variant of the

Kakeya set In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional ...

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. These conjectures were solved by Bourgain.

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..

References

{{reflist Measure theory Dimension theory Metric geometry Conjectures Unsolved problems in geometry