A construction of a $λ$- Poisson generic sequence
Verónica Becher, Gabriel Sac Himelfarb
TL;DR
The paper addresses the explicit construction of $λ$-Poisson generic sequences over alphabets of size $b≥2$, extending the probabilistic fact that almost all sequences are Poisson generic to an explicit computable instance. The authors build sequences by concatenating blocks from an infinite de Bruijn sequence and distributing block repetitions according to a probability vector $(p_i)$ with $∑ p_i=1$ and $∑ i p_i=λ$, ensuring $Z^{λ}_{i,k}(x)→p_i$ for all $i$. This yields explicit $λ$-Poisson generic sequences for all $b≥2$ and $λ>0$ with a binary restriction $λ≤ln 2$, and, as a corollary, proves Borel normality; the work also shows that normal sequences can fail to be $λ$-Poisson generic. Additionally, a norm criterion (inspired by Pyatetskii-Shapiro) shows that if $p_i=liminf Z^{λ}_{i,k}(x)$ and $∑ i p_i=λ$, then the sequence is Borel normal, clarifying the relationship between Poisson genericity and normality. The construction is computable when $(p_i)$ is computable, providing explicit computable instances of $λ$-Poisson generic sequences and highlighting a rich interaction between de Bruijn structures and Poisson-type word frequencies.
Abstract
Years ago Zeev Rudnick defined the $λ$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $λ$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $λ$-Poisson generic sequence over any alphabet and any positive $λ$, except for the case of the two-symbol alphabet, in which it is required that $λ$ be less than or equal to the natural logarithm of $2$. Since $λ$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $λ$-Poisson generic.