Consider a sum of exponentials function of the integer $x$: $f(x)=A(x)+B(x)$, where $A(x)=\sum_{i=1}^n c_i \theta_i^x$ with $\theta_i$ roots of unity, and $B(x)=\sum_{j=1}^m d_j\lambda_j^x$ with $|\lambda_j|=1$ but not roots of unity. Assume the $\theta_i$'s and $\lambda_j$'s are all distinct, and that the coefficients $c_i,d_j$'s are all nonzero. It is known that $A(x)$ is a periodic function of period say $T_A$.
Now let us assume that $g(x):=|f(x)|$ is periodic of period $T_g$.
Question: what can be said about the relation between $T_g$ and $T_A$? In particular, is it true that $T_g\geq T_A$? Any reference would be appreciated.
