When to answer and when not to answer a brief e-mail from a busy academic? I frequently receive brief, usually confirmatory e-mails (such as "Can do. Best" in response to a request for a letter of recommendation) from academics with considerably less time than myself.

Each time this happens, I am very indecisive about whether the nuisance caused by responding -- and adding an e-mail with essentially no information, such as "Many thanks; this means a lot to me! All the best" to their inbox -- really weighs more heavily than the risk of being perceived as ungrateful.

Any thoughts, ideally from the referee's perspective are greatly appreciated!
My own busy-person email triage has three tiers:


No response needed = near-zero cost
Minimal response needed = minimal cost
Response with significant thought and care needed = significant context-switching cost


A brief "thank you" email would go into category #1: no bother or burden, but noticed and appreciated all the same.  In other words, send the email: it's not a significant nuisance, and it's nice to be appreciated.