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The hypergraph regularity method and hypergraph removal lemma can prove high-dimensional and ring analogues of density version of Szemerédi's theorems, originally proved by Furstenberg and Katznelson. In fact, this approach yields first quantitative bounds for the theorems.
This theorem roughly implies that any dense subset of <math> \mathbb{Z}^d </math> contains any finite pattern of <math> \mathbb{Z}^d </math>. The case when <math> d = 1 </math> and the pattern is arithmetic progression of length some length is equivalent to Szemerédi's theorem.<blockquote>
==== Furstenberg and Katznelson Theorem ====
Let <math> T </math> be a finite subset of <math> \mathbb{R}^d </math> and let <math> \delta > 0 </math> be given. Then there exists a finite subset <math> C \subset \mathbb{R}^d </math> such that all <math> Z \subset C </math> with <math> |Z| > \delta |C| </math> contains a homothetic copy of <math> T </math>. (i.e. set of form <math> z + \lambda T </math>, for some <math> z \in \mathbb{R}^d </math> and <math> t \in \mathbb{R} </math>)
Moreover, if <math> T \subset [-t; t]^d </math> for some <math> t \in \mathbb{N} </math>, then there exists <math> N_0 \in \mathbb{N} </math> such that <math> C = [-N,N]^d </math> has this property for all <math> N \geq N_0 </math>.</blockquote>Another possible generalization is when it is allowed the dimension to grow.<blockquote>
==== Tengan, Tokushige, V.R., and M.S Theorem ====
Let <math> A </math> be a finite ring. For every <math> \delta > 0 </math>, there exists <math> M_0 </math> such that, for <math> M \geq M_0 </math>, any subset <math> Z \subset A^M </math> with <math> |Z| > \delta |A^M| </math> contains a coset of an isomorphic copy of <math> A </math> (as a left <math> A </math>-module).
In other words, there are some <math> \mathbf{r}, \mathbf{u} \in A^M </math> such that <math> r + \varphi(A) \subset Z </math>, where <math> \varphi \colon A \to A^M </math>, <math> \varphi(a)=a \mathbf{u} </math> is an injection.
\end{theorem}</blockquote>
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