The election for president is very close in Georgia.
What was the smallest margin of victory in both percentage points and raw votes for a presidential election in the US state of Georgia?
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I will base my answers on the tables on Wikipedia listing presidential election results in the state of Georgia.
Going by victory margin in percentage points, the closest race was in 1992. Bill Clinton received 43.47% of the vote and George H. W. Bush received 42.88% of the vote. That means the margin of victory in percentage points was 0.59%.
The absolute margin is probably not very interesting, because early on the population was a lot smaller and it hasn't had universal suffrage for a long time, and as such the margins in absolute numbers were often smaller as well. In 1836, Martin Van Buren received 22,778 votes and Hugh Lawson White received 24,481 votes. That means the margin of victory in votes was 1,703 votes.