Partitions by congruent sets and optimal positions

2010 Ergodic Theory and Dynamical Systems  
Let X be a metrizable space with a continuous group or semi-group action G. Let D be a nonempty subset of X. Our problem is how to choose a fixed number of sets in {g −1 D; g ∈ G}, say σ −1 D with σ ∈ τ , to maximize the cardinality of the partition P An infinite subset Σ of G is called an optimal position of the triple (X, G, D) if #P({σ −1 D; σ ∈ τ }) = p * X,G,D (k), holds for any k = 1, 2, · · · and τ ⊂ Σ with #τ = k. In this paper, we discuss examples of the triple (X, G, D) admitting or
more » ... t admitting an optimal position. Let X = G = R n (n ≥ 1), where the action g ∈ G to x ∈ X is the translation x − g. If D is the n-dimensional unit ball, then
doi:10.1017/s0143385709001175 fatcat:uuoj5njebfgflhlifzu5jqj76y