Normal numbers and normality measure

TL;DR

This paper improves bounds on the minimal normality measure of binary sequences, showing it grows at most logarithmically squared, thus advancing understanding of pseudorandomness and disproving a previous conjecture.

Contribution

It establishes a tighter upper bound of order $( ext{log } N)^2$ for the minimal normality measure, resolving an open problem and disproving a conjecture.

Findings

The minimal normality measure is bounded above by $c ( ext{log } N)^2$.

The result disproves a conjecture by Alon et al.

The proof links normality measure to discrepancy of normal numbers.

Abstract

The normality measure $\mathcal{N}$ has been introduced by Mauduit and S{\'a}rk{\"o}zy in order to describe the pseudorandomness properties of finite binary sequences. Alon, Kohayakawa, Mauduit, Moreira and R{\"o}dl proved that the minimal possible value of the normality measure of an $N$-element binary sequence satisfies $$ (1/2 + o(1)) \log_2 N \leq \min_{E_N \in \{0,1\}^N} \mathcal{N}(E_N) \leq 3 N^{1/3} (\log N)^{2/3} $$ for sufficiently large $N$. In the present paper we improve the upper bound to $c (\log N)^2$ for some constant $c$, by this means solving the problem of the asymptotic order of the minimal value of the normality measure up to a logarithmic factor, and disproving a conjecture of Alon \emph{et al.}. The proof is based on relating the normality measure of binary sequences to the discrepancy of normal numbers in base 2.