# algebraic numbers with small norms

Does there exist an algebraic number $$\alpha$$ such that $$\left|\frac{\alpha^n+\alpha^n_1}{n!}\right|\sim_{n\to+\infty}\frac1{(n!)^2}$$ where $$\alpha_1$$ is a conjugate of $$\alpha$$? Obviously $$\alpha$$ can not be a rational number.

For any fixed nonzero complex numbers $$z_1,\dotsc,z_m$$, there are infinitely many $$n$$'s such that the arguments of $$z_1^n,\dotsc,z_m^n$$ all lie in $$[-\pi/4,\pi/4]$$. This follows from Dirichlet's theorem on simultaneous diophantine approximation. For such $$n$$'s, $$|z_1^n+\dotsb+z_m^n|\geq\Re(z_1^n+\dotsb+z_m^n)\geq\frac{|z_1|^n+\dotsb+|z_m|^n}{\sqrt{2}}.$$ In particular, the left hand side cannot be asymptotically $$1/n!$$, because the right hand side is exponentially small at worst.
In short, there is no $$\alpha$$ satisfying the requirements.