Decomposing the real line into Borel sets closed under addition |
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Authors: | Tamás Keleti |
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Affiliation: | 1. Alfréd Rényi Institute of Mathematics, Budapest, Hungary;2. Institute of Mathematics, E?tv?s Loránd University, Budapest, Hungary |
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Abstract: | We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on. |
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