Can polynomial-exponential sums have an infinite number of zeros in an interval? Let $$p(t) = \sum_{i=1}^n a_i t^{b_i} e^{c_i t}.$$ Suppose $p(t)$ is not constant. Could $p(t)$ have infinitely many zeros in an interval of finite length?

Indeed all the functions $p(t)$ are analytic, since they are linear combination of analytic functions. we will prove that there exists no analytic function such that
$$f:\mathbb R \to \mathbb R:\ \exists \{a_n\}_n\ f(a_n)=0,\ a_n\in [a,b]\ \forall n$$
since $[a,b]$ is compact, the sequence $a_n$ admits a sequence $b_n\to x\in [a,b]$.
supposing by contradiction that $f$ is analytic you find that
$$f(x)=\lim_n f(b_n)=0$$
moreover, you can use Rolle's theorem to repeat the argument on $f^{(n)}$ and show that every derivative of $f$ takes value $0$ in $x$. This means by analitycness that $f=0$ everywhere contradicting your assumption
