In mathematics, the Erdős–Ulam problem asks whether the plane contains a
dense set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
of points whose
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 are all
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s. It is named after
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 ...
and
Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
.
Large point sets with rational distances
The
Erdős–Anning theorem
The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1 ...
states that a set of points with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, let ''S'' be the set of points
:
where
is restricted to values that cause
to be a rational number. For each such point, both
and
are themselves both rational, and if
and
define two points in ''S'', then their distance is the rational number
:
More generally, a circle with radius
contains a dense set of points at rational distances to each other if and only if
is rational.
[.] However, these sets are only dense on their circle, not dense on the whole plane.
History and partial results
In 1946,
Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
asked whether there exists a set of points at rational distances from each other that forms a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of 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 ...
.
While the answer to this question is still open,
József Solymosi
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 ...
and Frank de Zeeuw showed that the only irreducible
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s that contain infinitely many points at rational distances are lines and circles.
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
and Jafar Shaffaf independently observed that, if the
Bombieri–Lang conjecture In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type.
Statement
The weak Bombie ...
is true, the same methods would show that there is no infinite dense set of points at rational distances in the plane.
Using different methods,
Hector Pasten
In Greek mythology, Hector (; grc, Ἕκτωρ, Hektōr, label=none, ) is a character in Homer's Iliad. He was a Trojan prince and the greatest warrior for Troy during the Trojan War. Hector led the Trojans and their allies in the defense o ...
proved that the
abc conjecture
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b' ...
also implies a negative solution to the Erdős–Ulam problem.
[
]
Consequences
If the Erdős–Ulam problem has a positive solution, it would provide a counterexample to the Bombieri–Lang
[ conjecture and to the ]abc conjecture
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b' ...
.[ It would also solve ]Harborth's conjecture
In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length.. This conjecture is named after Heiko Harborth, and (if true) would strengthen Fáry's theorem ...
, on the existence of drawings of planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
s in which all distances are integers. If a dense rational-distance set exists, any straight-line drawing of a planar graph could be perturbed by a small amount (without introducing crossings) to use points from this set as its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven.
References
{{DEFAULTSORT:Erdos-Ulam problem
Arithmetic problems of plane geometry
Unsolved problems in mathematics
Ulam problem