About 4.55pm on July 6, a building was set alight within the Keir Homes development at the former officers mess at the Barracks, Quinn Court, Lanark.
Anyone with information on who was responsible can contact Police Scotland 101, or Crimestoppers on 0800555111, where those with information will be treated in confidence, and can remain anonymous.
The police are also looking for information on other incidents.
Between July 4 and July 5 a housebreaking occurred in the Clyde Valley Country Estate, Crossford, and two flat screen TVs were stolen. Witnesses are sought.
There has also been a spate of break-ins to sheds including a door being removed from a shed at an address on Thomson Street Carluke on Sunday, July 10, and a door removed from a shed on Goremire Road, Carluke the same day. No property was stolen in either on these incidents.
But a strimmer was stolen from a shed on Murray Terrace, Carluke. That theft was discovered on July 10.