mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

combinatorial 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 geo ...

, the Littlewood–Offord problem is the problem of determining the number of subsums of a set of

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s that fall in a given

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

. More formally, if ''V'' is a vector space of

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

''d'', the problem is to determine, given a finite subset of vectors ''S'' and a convex subset ''A'', the number of subsets of ''S'' whose summation is in ''A''. The first

upper 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 greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...

for this problem was proven (for ''d'' = 1 and ''d'' = 2) in 1938 by

John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...

and A. Cyril Offord. This Littlewood–Offord lemma states that if ''S'' is a set of ''n'' real or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s of absolute value at least one and ''A'' is any disc of

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

one, then not more than

\Big( c \, \log n / \sqrt \Big) \, 2^n

of the 2^''n'' possible subsums of ''S'' fall into the disc. In 1945 Paul Erdős improved the upper bound for ''d'' = 1 to :

\approx 2^n \, \frac

using Sperner's theorem. This bound is sharp; equality is attained when all vectors in ''S'' are equal. In 1966, Kleitman showed that the same bound held for complex numbers. In 1970, he extended this to the setting when ''V'' is a normed space. Suppose ''S'' = . By subtracting :

\frac \sum_^n v_i

from each possible subsum (that is, by changing the origin and then scaling by a factor of 2), the Littlewood–Offord problem is equivalent to the problem of determining the number of sums of the form :

\sum_^n \varepsilon_i v_i

that fall in the target set ''A'', where

\varepsilon_i

takes the value 1 or −1. This makes the problem into a probabilistic one, in which the question is of the distribution of these ''

random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...

s'', and what can be said knowing nothing more about the ''v''_''i''.

References

Combinatorics Probability problems Lemmas Mathematical problems {{combin-stub