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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 ====
Source:<ref name=":0" /> 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 every <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 that can be proven by the removal lemma is when the dimension is allowed to grow.<blockquote>
==== Tengan, Tokushige, Rödl, and Schacht 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).
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