Periodic properties of coloured sets (Q2707268)

Let \(f:\mathbb R^n \to \mathbb R^1\) be a non-negative bounded function vanishing outside a bounded set. If \(f\) has got a finite range \(\{0,c_1,\ldots,c_m\}\), such that \(c_i > 0\) and the sets \(A_i := \{x \in \mathbb R^n : f(x)=c_i\}\) are non-empty and bounded, \(i = 1, 2, \ldots, m\), then \(f\) is called a coloured set and \(c_i\) its colours. Let \(L \subset \mathbb R^n\) be a point lattice of full dimension. The following question is studied: under what conditions can the coloured set \(f\) be extended periodically \(\pmod L\) to the whole space \(\mathbb R^n\)? For general \(f\) the author proves some sufficient and (or) necessary conditions for its periodic extendability. He finds five conditions that are equivalent to the periodic extendability of a coloured set. NEWLINENEWLINENEWLINEThe results of this paper both extend some of those proved by the author [Linear Algebra Appl. 241-243, 851-876 (1996; Zbl 0860.11034)], for one colour sets and refine some of those by him [e.g., The inner periodic structure of a function, Research Report, IRL-3, 1-21 (1995)], for general real valued bounded functions. NEWLINENEWLINENEWLINEThe results of the above mentioned papers seems to be interesting for the following reasons. Firstly, they yield substantial improvements of some basic results in the geometry of numbers, e.g., the theorem of Minkowski-Blichfeldt-Van der Corput, or the theorem of Siegel-Bombieri. Secondly, in the first of the papers a new connection of the results with partitions of a bounded set in \(\mathbb R^n\) has been discovered, which connection might contribute to new developments in both fields (the theory of partitions and the geometry of numbers, respectively). Thirdly, for the proofs, new results in the field of multidimensional Fourier analysis also had to be proved.