A proof that the square root of s for s not a perfect square is simply normal to base 2

TL;DR

This paper proves that square roots of non-perfect squares are simply normal to base 2, meaning their binary digit sequences have a limiting frequency of 1/2, using an elementary and self-contained proof.

Contribution

It provides the first elementary proof that square roots of non-perfect squares are simply normal to base 2, addressing a long-standing open problem.

Findings

Square roots of non-perfect squares are simply normal to base 2.

The proof is elementary and relies on the concept of tails of expansions.

Convergence of digit frequency to 1/2 is established for these numbers.

Abstract

Since E. Borel proved in 1909 that almost all real numbers with respect to Lebesgue measure are normal to all bases, an open problem has been whether simple irrationals like square root of 2 are normal to any base. We show that each number of the form square root of s for s not a perfect square is simply normal to base 2, that is, the averages of the first n digits of its dyadic expansion converge to 1/2. The proof is mostly elementary and self contained but some basic probability is used. The main idea centers on the notion of tails of an expansion, that is, the sequence of digits with index larger than any fixed integer n.