On the binary digits of $\sqrt{2}$

TL;DR

This paper investigates the distribution of binary digits in the expansion of √2, establishing lower bounds on the count of ones and demonstrating infinitely often improved bounds.

Contribution

It provides new lower bounds on the number of ones in the binary expansion of √2 and shows these bounds can be improved infinitely often.

Findings

Number of 1's in first N digits ≥ √2N(1+o(1))

Improved bounds around 2√N/√(2√2−1) occur infinitely often

Establishes distribution properties of binary digits of √2

Abstract

We show that the number of $1$'s in the first $N$ digits of the binary expansion of $\sqrt{2}$ is at least $\sqrt{2N}(1+o(1))$ and show that this bound can be improved to around $2\sqrt{N}/\sqrt{2\sqrt{2}-1}$ infinitely often.