''The Erdős Distance Problem'' is a

monograph A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...

on 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. ...

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geom ...

: how can one place

n

points into

d

-dimensional

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

so that the pairs of points make the smallest possible

distance set In geometry, the distance set of a collection of points is the Set (mathematics), set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a Minkowski difference, difference set, the set of distances (and th ...

? It was written by Julia Garibaldi, Alex Iosevich, and Steven Senger, and published in 2011 by the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

as volume 56 of the Student Mathematical Library. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

has suggested its inclusion in undergraduate mathematics libraries.

Topics

''The Erdős Distance Problem'' consists of twelve chapters and three appendices. After an introductory chapter describing the formulation of the problem by

Paul Erdős Paul Erdős ( ; 26March 191320September 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 discrete mathematics, g ...

and Erdős's proof that the number of distances is always at least proportional to

\sqrt /math>, the next six chapters cover the two-dimensional version of the problem. They build on each other to describe successive improvements to the known results on the problem, reaching a

lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less th ...

proportional to

n^

in Chapter 7. These results connect the problem to other topics including the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

, the crossing number inequality, the

Szemerédi–Trotter theorem The Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry. It asserts that given points and lines in the Euclidean plane, the number of incidences (''i.e.'', the number of point-line pairs, such that the point ...

on incidences between points and lines, and methods from

information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...

. Subsequent chapters discuss variations of the problem: higher dimensions, other

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s for the plane, the number of distinct

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

s between vectors, and analogous problems in spaces whose coordinates come from a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

instead of the

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

Audience and reception

Although the book is largely self-contained, it assumes a level of mathematical sophistication aimed at advanced university-level mathematics students. Exercises are included, making it possible to use it as a textbook for a specialized course. Reviewer Michael Weiss suggests that the book is less successful than its authors hoped at reaching "readers at different levels of mathematical experience": the density of some of its material, needed to cover that material thoroughly, is incompatible with accessibility to beginning mathematicians. Weiss also complains about some minor mathematical errors in the book, which however do not interfere with its overall content. Much of the book's content, on the two-dimensional version of the problem, was made obsolete soon after its publication by new results of

Larry Guth Lawrence David Guth (; born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale University in 2000 with a BS in mathematics. In 2005, he received his PhD in mathemati ...

and

Nets Katz Nets Hawk Katz is the W.L. Moody Professor of Mathematics at Rice University. He was a professor of mathematics at Indiana University Bloomington until March 2013 and the IBM Professor of Mathematics at the California Institute of Technology until ...

, who proved that the number of distances in this case must be near-linear. Nevertheless, reviewer

William Gasarch William Ian Gasarch ( ; born 1959) is an American computer scientist known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor at the University of M ...

argues that this outcome should make the book more interesting to readers, not less, because it helps explain the barriers that Guth and Katz had to overcome in proving their result. Additionally, the techniques that the book describes have many uses in discrete geometry.

References

{{DEFAULTSORT:Erdős Distance Problem Mathematics textbooks 2011 non-fiction books Discrete geometry American Mathematical Society