In all cases where, subsequent to the time provided by law for persons to claim lots on such town site, any person may have entered thereon and improved any lots belonging to such town, such person, after the report of said board of appraisers, and prior to public sale, may purchase any such lots from the judge or corporate authorities of such town for cash, at the appraised values of such lots, pieces or parcels of land, inclusive of improvements, unless there shall be adverse claimants to any such lots, in which case the respective rights of such claimants shall be determined as hereinafter provided.