Three-term Arithmetic Progression Research Articles

A Szemerédi-type regularity lemma in abelian groups, with applications

Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If $$A \subseteq \{1,\ldots,N\}$$ has δ N2 triples (a1, a2, a3) for which a1 + a2 = a3 then A = B ∪ C, where B is sum-free and |C| = δ′N, and $$\delta^{\prime} \rightarrow 0$$ as $$\delta \rightarrow 0.$$ Another answers a question of Bergelson, Host and Kra. If $$\alpha, \epsilon > 0,$$ if $$N\,>\,N_{0}(\alpha, \epsilon)$$ and if $$A \subseteq \{1,\ldots,N\}$$ has size α N, then there is some d ≠ 0 such that A contains at least $$(\alpha^{3}-\epsilon)N$$ three-term arithmetic progressions with common difference d.

On Rainbow Arithmetic Progressions

Consider natural numbers $\{1, \cdots, n\}$ colored in three colors. We prove that if each color appears on at least $(n+4)/6$ numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungić et al.

Matrix multiplication via arithmetic progressions

We present a new method for accelerating matrix multiplication asymptotically. Thiswork builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.