Verónica Becher

We define Poisson genericity for infinite sequences in any countable alphabet with an invariant exponentially ψ \psi -mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss’ theorem about Poisson genericity of integer bases numeration systems. In particular, we obtain that the continued fraction expansions of almost all real numbers are Poisson generic.

We introduce two-dimensional periodic arrays that are a variant of the de Bruijn tori. We call them nested perfect arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for n being a power of 2, we partition the positions of the arrays in n ² many congruence classes by taking the row number modulo n and the column number modulo n . We construct arrays where each possible n × n array occurs n ² times, once in each congruence class. Our method yields exponentially many (in n ² ) different nested perfect arrays.