Based on the original file, you can know what permissions you need. For the new file, fiddle around with File.setReadable(boolean readable), setWriteable(), setExecutable() API's to set the permission for the new files to be the exact same as the one for your original file.
EDIT:
NOTE. This is for Java >= 7.
To set all perms for a file (akin to chmod), move over to Files. There's this method setPosixFilePermissions(..). You will need to import the 3 relevant classes from java.nio.file.*
I had used it like so...
Set<PosixFilePermission> filePermission = new HashSet<PosixFilePermission>();
// OWNER rwe --- ---
filePermission.add(PosixFilePermission.OWNER_READ);
filePermission.add(PosixFilePermission.OWNER_WRITE);
filePermission.add(PosixFilePermission.OWNER_EXECUTE);
// GROUP --- rwe ---
filePermission.add(PosixFilePermission.GROUP_READ);
filePermission.add(PosixFilePermission.GROUP_WRITE);
filePermission.add(PosixFilePermission.GROUP_EXECUTE);
// GLOBAL --- --- rwe
filePermission.add(PosixFilePermission.OTHERS_READ);
filePermission.add(PosixFilePermission.OTHERS_WRITE);
filePermission.add(PosixFilePermission.OTHERS_EXECUTE);
Files.setPosixFilePermissions(Paths.get("{PATH_TO_FILE}"), filePermission);