A state oversight board voted on Wednesday to approve the Atlantic Yards project near Downtown Brooklyn, knocking down the last regulatory hurdle for the biggest real estate project in Brooklyn history.

The vote by the Public Authorities Control Board capped three years of bitter battles between opponents and supporters of the $4 billion project. The version approved today — 8 million square feet over 22 acres along Atlantic Avenue — includes a huge residential housing complex with about 6,400 market-rate and subsidized apartments, a basketball arena for the New Jersey Nets, and a smattering of office space, with the design punctuated by elaborately designed towers that dwarf nearby residential neighborhoods.

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