Liouville Random functions and normal sets

Abstract

We define a random Liouville function (\lambda_Q) which depends on a random set (Q) of primes and prove that (A_Q = \{n \in \mathbb{N} | \lambda_Q(n) = -1 \}) is normal almost everywhere. This fact enables us to generate a family of normal sets such that the equation (xy =z) is not solvable inside them. Additionally we prove that equations (xy=z^2, x^2 + y^2 = square, x^2 - y^2 = square) are solvable in any normal set and for any equation (xy=cn^2) ((c > 1 ), is not a square) there exists a normal set (A_c) such that the equation is not solvable inside (A_c).