Linux – Case Insensitive search from find command

NTFS¹ and VFAT are not case-sensitive, they are just case-preserving. That means if you create a file named FileName.txt, the file system will preserve the mixed case name, but you can access the file with whatever case combination of the same letters, like FILENAME.TXT, filename.txt or fileNAME.txt. This explains you cannot have two files with the same spelling with only a variation of upper/lower case in the same directory.

SMB exported file system have to implement this behavior not to confuse Windows clients.

ZFS can be configured to behave that way with the casesensitivity=mixed property.

¹ _{Technically, NTFS is case sensitive but the OSes mounting file systems of this type are almost always configured to hide this underlying feature and only preserve the case. Windows can however enable case sensitivity with modifying this register key HKLM\SYSTEM\CurrentControlSet\Control\Session Manager\kernel\dword:ObCaseInsensitive and Linux can mount these file systems with various behaviors depending on the ignore_case and windows_names mount options.}

find . -iname "*$(date +%d)*" -print

This should do the trick. With $() You can get the output of a command as an argument to another command.