A construction of a $\lambda$- Poisson generic sequence

TL;DR

This paper constructs explicit examples of $mbda$-Poisson generic sequences over any alphabet (except binary with certain conditions), demonstrating their properties and relation to Borel normality, filling a gap in explicit sequence examples.

Contribution

It provides the first explicit construction of $mbda$-Poisson generic sequences for any alphabet and positive $mbda$, except for binary cases with restrictions.

Findings

Constructed explicit $mbda$-Poisson generic sequences for any alphabet (except binary with $mbda > \, ext{ln}(2)$).

Showed these sequences are Borel normal.

Provided sequences that are Borel normal but not $mbda$-Poisson generic.

Abstract

Years ago Zeev Rudnick defined the ${\lambda}$-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 ${\lambda}$. 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 ${\lambda}$-Poisson generic sequence over any alphabet and any positive ${\lambda}$, except for the case of the two-symbol alphabet, in which it is required that ${\lambda}$ be less than or equal to the natural logarithm of $2$. Since ${\lambda}$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not ${\lambda}$-Poisson generic.