\(k\)-point configurations in sets of positive density of \(\mathbb{Z}^n\). (English) Zbl 1165.05029

Magyar, Ákos

Duke Math. J. 146, No. 1, 1-34 (2009).

Let \(\Delta\) denote \(k+1\) points in general position, i.e. a nondegenerate \(k\)-dimensional simplex.
J. Bourgain proved [”A Szemerédi type theorem for sets of positive density in \(R^ k\),” Isr. J. Math. 54, 307–316 (1986; Zbl 0609.10043)]: let \(E \subset \mathbb{R}^n\) be a set of positive upper density, and \(k<n\), then \(E\) contains a translated and rotated image of all large dilates of \(\Delta\).
In this paper the author proves a related result of embeddings into lattice points. Let \(\Delta=\{v_0, \ldots, v_k\}\subset\mathbb{R}^n\). \(\Delta\) is called integral, if all dot products \((v_i-v_0)\cdot (v_j-v_0)\) are integers. Let \(k \geq 2\), and let \(n>2k+4\). For each \(A \subseteq \mathbb{Z}^n\) with \(\delta(A)=\delta>0\), the following holds for all integral \(k\)-dimensional simplices \(\Delta\). There is a positive integer \(Q=Q(\delta)\) and a number \(\Lambda=\Lambda(A, \Delta)\) so that for all integers \(\lambda>\Lambda\) there is a simplex \(\Delta' \subseteq A\) which is, up to a rigid motion, \(\sqrt{\lambda} Q \Delta\). Further results contain a quantitative version.
The existence of a dilate \(\lambda \Delta\) can also be proved from a multidimensional version of Szemerédi’s theorem. But this would not achieve the same type of quantitative versions. Moreover, in a sense the author proves the existence of all dilates in \(A\).
The methods involve Fourier analysis and a version of the circle method, due to Siegel.

Reviewer: Christian Elsholtz (Surrey)

MSC:

05D10	Ramsey theory
11B05	Density, gaps, topology
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)

Keywords:

geometric Ramsey theory

Citations:

Zbl 0609.10043

References:

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