clientModified & serverModified date

The Dropbox API doesn't offer a way to list/filter files only for a specific date, but I'll pass this along as a feature request. I can't promise if or when that might be implemented though. 

Instead, you'll need to list items using the /2/files/list_folder[/continue] endpoints via the process and pagination detailed in the documentation, iterate over the entries, and filter to the dates you want. Once you do that though, you can stay up to date with just new changes by storing the last cursor you received, and then continuing to call /2/files/list_folder/continue as necessary; that will return just new changes since you last called.

Note that the clientModified time is supplied by the client (e.g., official Dropbox desktop app, third party API app, etc.), so it should not be trusted or used for file change detection.