On the Number of Hecke Eigenvalues of Same Sign on ${\mathrm{G}\mathrm{L}}_n$

We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$ of self-dual Hecke–Maass cusp forms for the group ${\mathrm{S}\mathrm{L}}_n(\mathbb{Z})$, where $n\ge 2$ is an integer. Our main result establishes, under Generalised Ramanujan–Petersson Conjecture, that for almost all $x\sim X$ the short interval $[x,x+H]$ contains a subset $\mathcal{S}$ (resp. ${\mathcal{S}}^{\mathrm{\prime }})$ of size $\gg H(\log X{)}^{1\mathrm{/}{n}^2-1}$ such that $A(m,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$ is positive (resp. negative) for all $m\in \mathcal{S}$ (resp. $m\in {\mathcal{S}}^{\mathrm{\prime }}$), provided that $(\log X{)}^{1-1\mathrm{/}{n}^2}\ll H\ll X$. We also prove a slightly stronger result unconditionally for ${\mathrm{G}\mathrm{L}}_2$ and ${\mathrm{G}\mathrm{L}}_3$ Hecke–Maass cusp forms. In addition, we obtain results under weaker bounds towards Generalised Ramanujan–Petersson Conjecture. Finally, as a by-product of our methods, we improve earlier bounds for the number of Hecke eigenvalues of same sign also in long intervals unconditionally for ${\mathrm{G}\mathrm{L}}_2$ and ${\mathrm{G}\mathrm{L}}_3$ Hecke–Maass cusp forms, and under Generalised Ramanujan–Petersson Conjecture for ${\mathrm{G}\mathrm{L}}_n$ forms when $n\ge 4$.