On almost $p$-rational characters of $p^{\mathrm{\prime }}$-degree

Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2\pi i\mathrm{/}n})$ for some $n\in {\mathbb{Z}}^{+}$ prime to $p$ or precisely divisible by $p$. We prove that, in contrast to usual $p$-rational characters, there are always “many” almost $p$-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost $p$-rational irreducible characters of $G$ in terms of $p$. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for the number of such characters of $p^{\mathrm{\prime }}$-degree and prove that, in the minimal situation, the number of almost $p$-rational irreducible $p^{\mathrm{\prime }}$-characters of $G$ coincides with that of $N_G(P)$ for $P\in {\mathrm{S}\mathrm{y}\mathrm{l}}_p(G)$. Lastly, we propose a new way to detect the cyclicity of Sylow $p$-subgroups of a finite group $G$ from its character table, using almost $p$-rational irreducible $p^{\mathrm{\prime }}$-characters and the blockwise refinement of the McKay-Navarro conjecture.