Roth's theorem

This article deals with the "Roth theorem" on arithmetic sequences, for the theorem on Diophantine approximation see Thue-Siegel-Roth theorem .

The set of Roth is a theorem from the mathematical field of number theory . It says that in certain subsets of the whole numbers there are infinitely many arithmetic sequences of length . It was later generalized by Szemerédi's theorem. ${\displaystyle 3}$

Roth's theorem

Let it be a subset of the whole numbers with positive upper density: ${\displaystyle A\subset \mathbb {Z} }$

${\displaystyle \lim _{N\to \infty }{\frac {\sharp (A\cap \left[-N,N\right])}{2N+1}}>0}$,

then there are infinitely many arithmetic sequences of length , i.e. of form ${\displaystyle A}$ ${\displaystyle 3}$

${\displaystyle \left\{a,a+r,a+2r\right\}\subset A}$

with . ${\displaystyle a\in \mathbb {Z} ,r\in \mathbb {N} _{>0}}$

variants

Let it be an odd number and . Then for each one , so that for all sets with the inequality ${\displaystyle N}$ ${\displaystyle G=\mathbb {Z} /N\mathbb {Z} }$ ${\displaystyle \alpha \in \left(0,1\right)}$ ${\displaystyle c_{\alpha }>0}$ ${\displaystyle A\subset \mathbb {Z} /N\mathbb {Z} }$ ${\displaystyle \sharp A\geq \alpha N}$

${\displaystyle \sharp \left\{x,r\in \mathbb {Z} /N\mathbb {Z} \colon x,x+r,x+2r\in A\right\}\geq c_{\alpha }N^{2}}$

applies.

This rate applies generally for 2-divisible groups: It is a compact 2-divisible Abelian group with Haarschem probability , then, for every one , so that for every measurable set with the inequality ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle \alpha \in \left(0,1\right)}$ ${\displaystyle c_{\alpha }>0}$ ${\displaystyle A\subset G}$ ${\displaystyle \mu (A)\geq \alpha }$

${\displaystyle \int _{G}\int _{G}1_{A}(x)1_{A}(x+r)1_{A}(x+2r)\,\mathrm {d} \mu (x)\mathrm {d} \mu (r)\geq c_{\alpha }}$

applies.

A stronger form is the Roth-Khintschin theorem .

literature

• Klaus Friedrich Roth : On certain sets of integers. J. London Math. Soc. 28, (1953). 104-109.

Web links

• A proof of Roth's theorem
• Roth's theorem
• Roth's Theorem on 3-term Arithmetic Progressions
• Roth's Theorem: the Fourier Analytic Approach

Individual evidence

1. ↑ Varnavides: On Certain sets of positive density. J. London Math. Soc. 34 1959 358-360.
2. ↑ Meshulam: On subsets of finite abelian groups with no 3-term arithmetic progressions. J. Combin. Theory Ser. A 71 (1995), no. 1, 168-172.