I would expect a percentage, and not a hard number.
A recount of the vote is required when the canvass of returns in a primary or general election shows that the margin between the two candidates receiving the greatest number of votes for a particular office, or between the number of votes cast for and against initiated or referred measures or proposals to amend the Constitution of Arizona, is less than or equal to the lesser of the following:
1. One-tenth of one per cent of the number of votes cast for both such candidates or upon such measures or proposals.
2. Two hundred votes in the case of an office to be filled by state electors and for which the total number of votes cast is more than twenty-five thousand.
3. Fifty votes in the case of an office to be filled by state electors and for which the total number of votes cast is twenty-five thousand or less.
4. Two hundred votes in the case of an initiated or referred measure or proposal to amend the constitution.
5. Fifty votes in the case of a member of the legislature.
6. Ten votes in the case of an office to be filled by the electors of a city or town or a county or subdivision of a city, town or county.
Just listed to Rep Biggs on WMAL. He was asked the question about an automatic recount. He said the differences would have to be around 4,000 votes, either way.