3 problems

Let $B_\sigma:[0,1]\to\mathbb R$ be the $\sigma$-Brjuno function, and let $\eta_n=(\sqrt{n^2+4}-n)/2$ be the $n$-th fixed point of the Gauss map. For each $n\in\mathbb N$, consider…

Let $B_1$ denote the classical Brjuno function, let $mathcal{M}$ be the set of its local minima, and let $mathcal{N}$ be the set of noble numbers, namely the numbers whose regular…

For even $k\in\mathbb{N}$, define … where $\sigma_{k-1}(n)=\sum_{d\mid n}d^{k-1}$. Let $p_n(x)/q_n(x)$ be the $n$th continued-fraction convergent of $x$, and call … the $k$-Brjuno…