With vice-lawman Páll Vídalín, he was assigned by the king to survey conditions in Iceland; this took ten years, from 1702 to 1712, most of which time he spent there. He returned to Copenhagen for two winters, the first time to present trade proposals, and during the second, which was in connection with a court case, he got married. The ultimate results of the survey were the Icelandic census of 1703 and the "Jarðabók" or land register, surveying for which was not completed until 1714 and which had to be translated into Danish after Árni's death. He was expected to translate it himself, but it was one of several official tasks he neglected. It was finally published in 11 volumes in 1911–41. However, the mission as set out by the king was extremely broad, including investigating the feasibility of sulphur mining, assessing the fisheries, and auditing the administration of justice. Complaints of judicial abuse poured in, and officials were incensed by the two men's inquiries into past court cases and in turn complained to Copenhagen about them.