You need to know: Set of vectors with integer coordinates, countable and uncountable sets, computable real number.
Background: Let be a finite set of points in , and let be a finite set of colours. Let L be any subset of colourings of F. A translate of F by is the set . A colouring of in colours from is called admissible for L if all translates of F are coloured according to a pattern belonging to L. The shift of finite type (SFT) defined by L is the set X of all admissible colourings of .
For every positive integer n, let be the set of points such that for all . For the SFT X, let be the number of different colourings of which can appear in some colouring . The (topological) entropy of X is then defined as .
The Theorem: On 7th March 2007, Michael Hochman and Tom Meyerovitch submitted to arxiv a paper in which they proved the existence of SFT X such that real number is not computable.
Short context: Because there are uncountably many real numbers and only countably many algorithms, there exist real numbers which are not computable. However, important real numbers naturally arising in mathematics and applications, such as and , are computable. The Theorem provides examples of real numbers which are not computable but quite meaningful, because SFTs are important objects of study in symbolic dynamics, and the entropy is their most studied invariant. The Theorem is a corollary of a more general result stating that a real number is the entropy of some SFT if and only if it is right recursively enumerable, that is, there is a computable sequence of rational numbers converging to h from above.
Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.9 of this book for an accessible description of the Theorem.
Theorems of the 21st century 