Falconer's conjecture

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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 -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 is a compact set of points in -dimensional Euclidean space whose Hausdorff dimension is strictly greater than , then the conjecture states that the set of distances between pairs of points in must have nonzero Lebesgue measure. [1]

Contents

Formulation and motivation

Falconer (1985) proved that Borel sets with Hausdorff dimension greater than have distance sets with nonzero measure. [2] 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 for some . [3] 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. [4]

Partial results

Erdoğan (2005) proved that compact sets of points whose Hausdorff dimension is greater than have distance sets with nonzero measure; for large values of this approximates the threshold on Hausdorff dimension given by the Falconer conjecture. [5] For points in the Euclidean plane, Borel sets of Hausdorff dimension greater than 5/4 (or with ) 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. [6] For the best known bound is according to a preprint by Du, Ou, Ren and Zhang [7] [8]

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. [9]

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. [10] 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. [11] [12]

Related Research Articles

In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

<span class="mw-page-title-main">Compact space</span> Type of mathematical space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number linewould be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

<span class="mw-page-title-main">Hausdorff dimension</span> Invariant measure of fractal dimension

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

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 higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).

<span class="mw-page-title-main">Metric space</span> Mathematical space with a notion of distance

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

<span class="mw-page-title-main">Null set</span> Measurable set whose measure is zero

In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.

<span class="mw-page-title-main">Diophantine approximation</span> Rational-number approximation of a real number

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume ,

In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.

<span class="mw-page-title-main">Kakeya set</span> Shape containing unit line segments in all directions

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 space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero.

<span class="mw-page-title-main">Minkowski–Bouligand dimension</span> Method of determining fractal dimension

In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand.

In mathematics, the Littlewood conjecture is an open problem in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,

In the mathematical discipline of complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.

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.

In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces. The term "shy" was suggested by the American mathematician John Milnor.

<span class="mw-page-title-main">Equilateral dimension</span> Max number of equidistant points in a metric space

In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of -dimensional Euclidean space is , achieved by a regular simplex, and the equilateral dimension of a -dimensional vector space with the Chebyshev distance is , achieved by a hypercube. However, the equilateral dimension of a space with the Manhattan distance is not known. Kusner's conjecture, named after Robert B. Kusner, states that it is exactly , achieved by a cross polytope.

<span class="mw-page-title-main">József Solymosi</span> Hungarian-Canadian mathematician

József Solymosi is a Hungarian-Canadian mathematician and a professor of mathematics at the University of British Columbia. His main research interests are arithmetic combinatorics, discrete geometry, graph theory, and combinatorial number theory.

In geometry, the distance set of a collection of points is the set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a difference set, the set of distances in collections of numbers.

<span class="mw-page-title-main">Blichfeldt's theorem</span> High-area shapes can shift to hold many grid points

Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area , it can be translated so that it includes at least points of the integer lattice. Equivalently, every bounded set of area contains a set of points whose coordinates all differ by integers.

References

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  3. Steinhaus, Hugo (1920), "Sur les distances des points dans les ensembles de mesure positive" (PDF), Fundamenta Mathematicae (in French), 1 (1): 93–104, doi: 10.4064/fm-1-1-93-104 .
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  8. Sloman, Leila (2024-04-09). "Number of Distances Separating Points Has a New Bound". Quanta Magazine. Retrieved 2024-04-10.
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