In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
Klaus Friedrich Roth was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.
In mathematics, a sum-free sequence is an increasing positive integer sequence
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group G and a positive integer n, one asks for the smallest value of k such that every sequence of elements of G of size k contains n terms that sum to 0.
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets A and B of elements from an abelian group G,
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of
In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a non-complete graph H. It is named after Paul Erdős and Arthur Stone, who proved it in 1946, and it has been described as the “fundamental theorem of extremal graph theory”.
In mathematics, the Hardy–Ramanujan theorem, proved by G. H. Hardy and Srinivasa Ramanujan (1917), states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log ).
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.
In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this number cannot be too close to being a linear function.
In number theory, a Sidon sequence, named after the Hungarian mathematician Simon Sidon, is a sequence A = {a0, a1, a2, ...} of natural numbers in which all pairwise sums ai + aj (i ≤ j) are different. Sidon introduced the concept in his investigations of Fourier series.
The Erdős–Turán conjecture is an old unsolved problem in additive number theory posed by Paul Erdős and Pál Turán in 1941.
In arithmetic combinatorics, the Erdős–Szemerédi theorem, proven by Paul Erdős and Endre Szemerédi in 1983, states that, for every finite set of real numbers, either the pairwise sums or the pairwise products of the numbers in the set form a significantly larger set. More precisely, it asserts the existence of positive constants c and such that
In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if is a sequence of i.i.d. random variables with zero mean and finite variance and
The Furstenberg–Sárközy theorem is a result in additive number theory on sets of numbers no two of which differ by a square. It states that, if is a set of natural numbers with the property that no two numbers in differ by a square number, then the natural density of is zero. That is, for every , and for all sufficiently large , the fraction of the numbers up to that are in is less than . Equivalently, every set of natural numbers with positive upper density contains two numbers whose difference is a square. The theorem is named after Hillel Furstenberg and András Sárközy, who proved it in the late 1970s.
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive basis of every order. More specifically, it states that for every fixed integer , there exists a subset of the natural numbers satisfying
Sequences is a mathematical monograph on integer sequences. It was written by Heini Halberstam and Klaus Roth, published in 1966 by the Clarendon Press, and republished in 1983 with minor corrections by Springer-Verlag. Although planned to be part of a two-volume set, the second volume was never published.
