Hi Are,
"Arns" is located some 2km north of the east end of Fannyside Loch, which is itself approx 3km east of Cumbernauld New Town. OS Sheet 65 (Falkirk and West Lothian) Grid Ref 756809. The 6" OS Map from 1843-1882 shows the site in fair detail. Check out the National Library of Scotland site (geo.nls.uk/search).
"The Mote" is shown on the same series of maps as being approx 1km. NE of Cumbernauld House. This however may just refer to an old, man-made earth feature, (cf "motte") and not a building. An enquiry to North Lanarkshire Council Library Service may provide more information about the area.
Good Luck,
Istrice