Pythagorean triples in level sets of completely multiplicative functions

Abstract

We show that given completely multiplicative functions $f_1,\dots,f_d$ taking values in the unit circle, there exist Pythagorean triples (i.e., integer solutions to $x^2+y^2=z^2$) with $f_i(x),f_i(y),f_i(z)$ all arbitrarily close to $1$ for all $i$. This is a new special case of the conjecture that any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple. Our proof combines vanishing averages for aperiodic functions with concentration estimates for pretentious functions. A similar proof is applied to obtain the analogous statement for more general equations of the form $ax^2+by^2=cz^2$ whenever $a,b,c$ are perfect squares satisfying the Rado's condition.