For the statement of Carlson's theorem please see,

https://en.wikipedia.org/wiki/Carlson%27s_theorem.

There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish on integers can be replaced with $f$ vanishing on a subset A of integers provided that $A$ has upper density 1. This is a necessary and sufficient condition.

Now my question is as follows: Assuming that the condition on the growth of $f$ on the $y$-axis is more stringent, say for example that $f$ is uniformly bounded on the entire $y$-axis, can one obtain an extension of Carlson's theorem with $f$ vanishing on a monotone divergent sequence $a_1<a_2<\ldots$ with upper density strictly less than one?