Jakub Konieczny

Combinatorial properties of Nil-Bohr sets

We explore connections between two classes of sets of integers. On one hand, we have the Nil-Bohr sets, which are the natural analogues of Bohr sets in higher order Fourier analysis, and arise from dynamics on nilmanifolds. On the other hand - we have the purely combinatorial property SG*, which is a slight modification of the better known IP*. It was first observed by Host and Kra that the two classes mentioned above are closely connected, in that every SG*-set is piecewise Nil-Bohr. We prove a partial converse to this statement, thus giving a partial combinatorial characterisation of the class of Nil-Bohr sets.